What is the Maximum Temperature for a Given Heat Capacity Equation?

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In summary, the conversation is about finding the temperature at which the heat capacity is maximum, using the equation exp(x) = (x + 2)/(x - 2) and solving for x. The person has calculated a numerical solution of x = 2.4 and the maximum heat capacity for this value agrees with their textbook. They were unsure if there was a different, "neater" solution, but it is confirmed that a numerical solution is valid in this case.
  • #1
jrevill
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Homework Statement



This regards a question on heat capacity; I'm trying to find the temperature at which the heat capacity is maximum. After differentiating the heat capacity expression, equating to zero, and rearranging (all of which I have omitted), my problem boils down to:

exp(x) = (x + 2)/(x - 2)

Solve for x. (x in this context is theta/T, but I don't think it's relevant since I'm trying to find their ratio)

Homework Equations



See above

The Attempt at a Solution



I calculated x = 2.4 by trial-and-error, and the maximum heat hapacity for this value is in agreement with the my textbook. I just don't know how to solve it properly!
 
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  • #2
What do you mean by "solve it properly"? A numerical solution is perfectly valid. In this equation, with x both in and outside of the exponential, there is no "elementary" solution. You could try Lambert's W function which is defined as the inverse of the function xex but that is no more "proper" than a numerical solution.
 
  • #3
Ah, sorry, I just thought there was a "neater" solution that was eluding me. Thanks for clearing that up.
 

1. What is an exponential equation?

An exponential equation is an equation in which the variable appears in an exponent. It typically takes the form of y = ab^x, where a and b are constants and x is the variable. These types of equations are commonly used in science and mathematics to model exponential growth or decay.

2. How do you solve an exponential equation?

To solve an exponential equation, you can use logarithms to isolate the variable. Take the logarithm of both sides of the equation and use the properties of logarithms to simplify. Then, solve for the variable. It is important to check for extraneous solutions and make sure any values are within the domain of the original equation.

3. What is the difference between a linear and exponential equation?

A linear equation has a variable that appears in the first power, while an exponential equation has a variable that appears in an exponent. This means that in a linear equation, the rate of change is constant, while in an exponential equation, the rate of change increases or decreases rapidly as the variable changes.

4. How are exponential equations used in the real world?

Exponential equations are used to model many real-world phenomena, such as population growth, compound interest, and radioactive decay. They can also be used to predict future values based on past data.

5. What is the inverse of an exponential equation?

The inverse of an exponential equation is a logarithmic equation. This means that if you switch the x and y values in an exponential equation, you can solve for the inverse function by taking the logarithm of both sides. The inverse function can also be used to find the input value that produces a specific output value in the original exponential equation.

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