Newton's Law of Cooling Calculus, Example Problems, Differential Equations

The Organic Chemistry Tutor

7 Feb 201723:05

EducationalLearning

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TLDRThe video script presents a comprehensive explanation of Newton's Law of Cooling, a principle that describes how the temperature of an object changes over time as it approaches the ambient temperature of its surroundings. The presenter walks through an example problem involving an object cooling from 100 degrees Celsius to 80 degrees Celsius in 12 minutes within a room at 50 degrees Celsius. By applying the law's formula, which incorporates the initial temperature (T_o), the surrounding temperature (T_s), and a constant (k), the video demonstrates how to calculate the time it takes for the object's temperature to decrease further to 70 degrees Celsius. The script also delves into deriving the formula, emphasizing the relationship between the rate of temperature change and the temperature difference between the object and its surroundings. The video concludes with a clear demonstration of how to apply Newton's Law of Cooling to solve practical problems, providing a solid foundation for understanding the concept.

Takeaways

📚 Newton's Law of Cooling describes how the temperature of an object changes over time as it approaches the ambient temperature.
⏱️ It takes 12 minutes for an object at 100°C to cool to 80°C in a room at 50°C.
🔢 The formula for temperature at any time t is T(t) = Ts + (To - Ts) * e^(-kt), where To is the initial temperature, Ts is the surrounding temperature, k is a cooling constant, and t is time.
🔍 To find the cooling constant k, use the initial temperature, final temperature after a certain time, and the surrounding temperature in the formula.
🧮 The value of k is calculated to be 0.04257 using the given times and temperatures in the problem.
⏳ It takes approximately 9.52 minutes for the object's temperature to decrease from 80°C to 70°C.
🌡️ The total time for the object's temperature to decrease from 100°C to 70°C is the sum of the individual cooling times, which is 21.5 minutes.
📉 The rate of temperature change decreases as the object's temperature gets closer to the surrounding temperature.
🚨 The derivative of temperature with respect to time is proportional to the negative difference between the object's temperature and the surrounding temperature, indicating the object is cooling.
∫ The formula for Newton's Law of Cooling is derived by integrating the differential equation representing the rate of temperature change over time.
📌 The constant c in the formula is found using the initial temperature of the object and represents the difference between the initial and surrounding temperatures.
📐 The final form of Newton's Law of Cooling is T(t) = Ts + (To - Ts) * e^(-kt), which can be used to solve for the temperature of an object at any time t.

Q & A

What is Newton's Law of Cooling?
-Newton's Law of Cooling is a principle that describes how the temperature of an object changes over time as it approaches the ambient temperature of its surroundings. It states that the rate of change of the temperature is proportional to the difference between the object's temperature and the surroundings.
What is the formula for Newton's Law of Cooling?
-The formula for Newton's Law of Cooling is T(t) = Ts + (To - Ts) * e^(-kt), where T(t) is the temperature of the object at time t, Ts is the surrounding temperature, To is the initial temperature of the object, k is the cooling constant, and e is the base of the natural logarithm.
How is the cooling constant (k) determined?
-The cooling constant (k) is determined by using the initial temperature of the object, the surrounding temperature, and the time it takes for the object to cool to a certain temperature. It is found by solving the equation for k after substituting known values.
What does the 'e' in the formula represent?
-In the context of Newton's Law of Cooling, 'e' represents the base of the natural logarithm, which is approximately equal to 2.71828. It is used in the formula to model the exponential decay of the temperature difference between the object and its surroundings.
How does the rate of temperature change relate to the temperature difference?
-The rate of temperature change is directly proportional to the temperature difference between the object and its surroundings. As the object's temperature gets closer to the surrounding temperature, the rate of cooling decreases, indicating a slower change in temperature.
What is the significance of the natural logarithm in deriving Newton's Law of Cooling?
-The natural logarithm is used to isolate the variable 'k' in the cooling equation. By taking the natural log of both sides of the equation, the exponent can be moved to the front, allowing for the calculation of the cooling constant 'k'.
How does the surrounding temperature affect the cooling process?
-The surrounding temperature (Ts) is a constant in the cooling equation and represents the ambient temperature of the environment. It affects the cooling process as the object's temperature (T(t)) will always be approaching this Ts value, with the rate of cooling depending on the difference between T(t) and Ts.
What happens when the object's temperature is lower than the surrounding temperature?
-When the object's temperature is lower than the surrounding temperature, heat will flow into the object, causing its temperature to increase. In this case, the rate of change of temperature with respect to time (dT/dt) will be positive, indicating a warming process.
How long does it take for an object at 100 degrees Celsius to cool to 70 degrees Celsius in a room at 50 degrees Celsius?
-It takes 21.5 minutes for an object at 100 degrees Celsius to cool to 70 degrees Celsius in a room at 50 degrees Celsius. This is derived from the sum of the time it takes to cool from 100 to 80 degrees Celsius (12 minutes) and from 80 to 70 degrees Celsius (9.5 minutes).
What is the relationship between the rate of temperature change and the difference in temperature between the object and its surroundings?
-The rate of temperature change is proportional to the negative difference in temperature between the object and its surroundings. This means that the greater the difference, the greater the rate of change, and vice versa. The negative sign indicates that the temperature of the object is decreasing when it is hotter than the surroundings.
How can the final temperature of an object be calculated using Newton's Law of Cooling?
-The final temperature of an object can be calculated by substituting the known values into the formula T(t) = Ts + (To - Ts) * e^(-kt) and solving for T(t). This will give the temperature of the object at any given time 't'.
What is the purpose of the constant of integration (c) in the derivation of Newton's Law of Cooling?
-The constant of integration (c) is used in the integration process to account for an undetermined constant value that arises when finding the antiderivative of a function. In the context of Newton's Law of Cooling, c is determined by the initial conditions, specifically the initial temperature of the object.
Why is the temperature of the surroundings considered a constant in Newton's Law of Cooling?
-The temperature of the surroundings is considered a constant because, in most practical scenarios, the room or environment's temperature remains relatively stable over the time scale of the cooling process being considered. This allows for a simplified model where Ts does not change.

Outlines

00:00

🔍 Introduction to Newton's Law of Cooling

This paragraph introduces Newton's Law of Cooling, which describes how the temperature of an object changes over time as it approaches the ambient temperature. The video presents an example problem where an object cools from 100°C to 80°C in 12 minutes in a room with a temperature of 50°C. The goal is to find out how much longer it will take for the object's temperature to decrease to 70°C. The paragraph explains the formula that will be used to solve this problem and defines the variables involved: the temperature of the surroundings (ts), the initial temperature of the object (to), and the temperature of the object at time t (t(t)). It also introduces the constant k, which is determined by solving an equation using the given cooling times and temperatures.

05:01

🧮 Calculating the Constant k and Solving the Problem

The paragraph walks through the process of calculating the constant k using the initial temperature of the object (100°C), the final temperature after 12 minutes (80°C), and the surrounding temperature (50°C). By solving the equation, the value of k is found to be 0.04257. With k determined, the video then shows how to use the formula to find the time it takes for the object's temperature to decrease from 80°C to 70°C. The process involves subtracting, dividing, and taking the natural logarithm of the relevant terms to solve for the new time (t), which is calculated to be approximately 9.5 minutes.

10:04

📉 Understanding the Rate of Temperature Change

This section discusses the concept of the rate of temperature change with respect to time. It explains that the rate of change is proportional to the difference in temperature between the object and its surroundings, as well as a constant k. The paragraph also touches on the negative sign in the rate of change equation, which indicates that the object is cooling. The video then derives the formula for Newton's Law of Cooling by integrating the rate of change equation and applying the initial conditions to solve for the constant of integration (c). The final derived formula is presented, showing how the temperature of the object (t(t)) changes over time.

15:05

🔧 Deriving Newton's Law of Cooling Formula

The paragraph provides an in-depth derivation of Newton's Law of Cooling formula. It starts by multiplying both sides of the rate of change equation by dt and then dividing by the difference between the object's temperature and the surrounding temperature. This leads to an expression that can be integrated. The integration process involves finding the antiderivative of the terms and adding a constant of integration. The natural logarithm is then used to simplify the equation, and algebraic manipulations are applied to isolate the temperature of the object at time t (t(t)). The initial temperature is used to solve for the constant c, which is then substituted back into the formula to give the final form of Newton's Law of Cooling.

20:05

📝 Summary and Conclusion

The final paragraph summarizes the process of deriving Newton's Law of Cooling and its application to the example problem. It reiterates the formula and explains how the initial temperature and the surrounding temperature, along with the constant k, are used to calculate the time it takes for an object to cool to a certain temperature. The video concludes by thanking the viewers for watching and wishing them a great day.

Mindmap

Keywords

💡Newton's Law of Cooling

Newton's Law of Cooling is a mathematical model that describes how the temperature of an object changes over time as it approaches the ambient temperature of its surroundings. In the video, this law is used to solve a problem involving the cooling of an object from 100 degrees Celsius to 70 degrees Celsius in a room at 50 degrees Celsius.

💡Temperature

Temperature is a physical quantity that expresses the thermal state of a body or environment. The video discusses the initial temperature of an object (100 degrees Celsius), the surrounding temperature (50 degrees Celsius), and the final temperatures (80 and 70 degrees Celsius) as key parameters in applying Newton's Law of Cooling.

💡Constant k

The constant k in Newton's Law of Cooling represents the rate at which an object cools. It is derived from the initial conditions of the cooling process. In the video, k is calculated to be 0.04257 using the time it takes for the object's temperature to decrease from 100 to 80 degrees Celsius.

💡Natural Log Function

The natural log function, or ln, is a mathematical function that is the inverse of the exponential function. In the context of the video, it is used to solve for the constant k and to find the time it takes for the object's temperature to decrease to a certain level, as it appears in the formula for Newton's Law of Cooling.

💡Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. In the video, the cooling of the object is described as an example of exponential decay, where the temperature of the object decreases more rapidly when the difference between its temperature and the surrounding temperature is larger.

💡Initial Temperature (T_o)

The initial temperature (T_o) is the starting temperature of the object at the beginning of the cooling process. In the video, the initial temperature is given as 100 degrees Celsius, which is the value used to begin the calculation for the cooling process.

💡Surrounding Temperature (T_s)

The surrounding temperature (T_s) is the temperature of the environment in which the object is placed. It remains constant throughout the cooling process. In the video, the surrounding temperature is consistently 50 degrees Celsius, serving as the baseline to which the object's temperature is cooling.

💡Final Temperature (T)

The final temperature (T) is the temperature of the object at a specific point in time during the cooling process. The video focuses on finding the time it takes to reach final temperatures of 80 and 70 degrees Celsius from an initial temperature of 100 degrees Celsius.

💡Time (t)

Time (t) is the independent variable in the context of Newton's Law of Cooling, representing the minutes or hours that have passed since the start of the cooling process. The video calculates the time it takes for the object to cool from 100 degrees Celsius to 80 degrees Celsius (12 minutes) and then to 70 degrees Celsius (9.5 minutes).

💡Derivative

In mathematics, a derivative represents the rate at which a quantity changes with respect to another quantity. In the video, the derivative of temperature with respect to time is used to express the rate of cooling, which is a key component in deriving Newton's Law of Cooling.

💡Integral

An integral is a mathematical concept that represents the accumulated quantity of a variable and is the reverse process of differentiation. In the video, integration is used to derive the formula for Newton's Law of Cooling by integrating the rate of temperature change with respect to time.

Highlights

Newton's Law of Cooling is the focus of the video, which explains how the temperature of an object decreases over time to match the ambient temperature.

An example problem is presented where it takes 12 minutes for an object at 100°C to cool to 80°C in a room at 50°C.

The formula for Newton's Law of Cooling is introduced, relating the temperature of an object at time 't' to the surrounding temperature, initial temperature, and a cooling constant 'k'.

The constant 'k' is determined by using the initial information about the object's temperature drop from 100°C to 80°C over 12 minutes.

The value of 'k' is calculated to be 0.04257 using the natural logarithm function.

The time taken for the object's temperature to decrease from 80°C to 70°C is found to be 9.5 minutes using the derived value of 'k'.

The total time to cool from 100°C to 70°C is confirmed to be 21.5 minutes by summing the individual time intervals and using the formula.

The rate of temperature change decreases as the object's temperature gets closer to the surrounding temperature.

The derivative of temperature with respect to time is shown to be proportional to the negative difference between the object's temperature and the surroundings.

The integration process to derive the formula for Newton's Law of Cooling is explained, involving the separation of variables and integration.

The constant of integration 'c' is determined using the initial temperature of the object.

The final form of Newton's Law of Cooling formula is presented, incorporating the surrounding temperature, the initial temperature difference, and the cooling constant 'k'.

The video emphasizes the practical application of Newton's Law of Cooling in solving real-world problems involving temperature changes.

The concept of heat flow from hotter to cooler objects is discussed, explaining why the temperature change rate is negative when the object is cooling.

The video concludes with a summary of how to use Newton's Law of Cooling to solve problems and a reminder of the importance of understanding the formula's derivation.

The importance of the difference in temperature between the object and surroundings on the rate of cooling is highlighted.

The video provides a step-by-step guide on how to derive and apply Newton's Law of Cooling formula, making it accessible for learners.

Transcripts

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