- #1

curious_mind

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- Homework Statement
- A box is divided into two equal halves with a partition. The volume of the entire box is ##V##. One partition contains the Real gas satisfying Dieterici Equation of state at temperature ##T_0##. Take Dieterici equation of state in this case as ## PV e^{\frac{a}{RTV}} = nRT ##

where ##a## is a constant. Now, the partition is removed instantaneously, and the gas is allowed to expand to fill the full volume of the box and come to equilibrium. Calculate the temperature of gas at the final stage when it is at equilibrium.

- Relevant Equations
- First internal energy equation, using Maxwell's relation ## \left( \dfrac{\partial U}{\partial V} \right)_T = T \left( \dfrac{\partial P}{\partial T} \right)_V - P ##.

I proceeded in the usual manner in which we take ##dU = 0## in the case of free expansion because there is no heat transfer in the box, as well as no work is done.

We can write, taking ## U ## as the function of ##V## and ##T##, $$ dU(V,T) = \left( \dfrac{\partial U}{\partial V} \right)_T ~ dV + \left( \dfrac{\partial U}{\partial T} \right)_V ~ dT $$ Also, ##\left( \dfrac{\partial U}{\partial T} \right)_V = C_V## is specific heat at constant volume.

Therefore, ## 0 = C_V ~ dT + \left( \dfrac{\partial U}{\partial V} \right)_T ~ dT ##.

Using the "First internal energy equation" given above, I calculated that (for Dieterici EoS, as given in the question) ## \left( \dfrac{\partial U}{\partial V} \right)_T = T \left( \dfrac{\partial P}{\partial T} \right)_V - P = \dfrac{Pa}{RTV}##.

[I have double verified my calculation, but still I request to verify that again.]

Using given equation of state, ## P = \dfrac{nRT}{V} e^{-\frac{a}{RTV}} ##. So, ## \left( \dfrac{\partial U}{\partial V} \right)_T = \dfrac{na}{V^2} e^{-\frac{a}{RTV}} ##

Therefore, I obtain the expression ## 0 = C_V ~ dT + \left( \dfrac{na}{V^2} e^{-\frac{a}{RTV}} \right) dV ##. Writing it in differential equation form,

$$ \dfrac{dV}{dT} = - \dfrac{C_V V^2}{na} e^{\frac{a}{RTV}} $$.

Now, I am not sure how to solve this differential equation, where I have to take limits for volume ##\frac{V}{2}## to ##V## and temperature from ##T_0## to temperature to be obtained ##T##.

Problem with the obtained differential equation is that, I am not getting it in a seperable form. Also, I know very little differential equation theory which could be helpful in solving this.I also tried Wolfram Mathemtica DSolve function, but it is also getting very large expressions.

The question is given here https://snboseplc.com/thermodynamics-tifr/ - Question number

Even if the answer is not accurate, how do I obtain the simplified answer ? How do people manage to get temperature change in Engineering thermodynamics to deal with such problems?Any help would be appreciable. Thanks.

We can write, taking ## U ## as the function of ##V## and ##T##, $$ dU(V,T) = \left( \dfrac{\partial U}{\partial V} \right)_T ~ dV + \left( \dfrac{\partial U}{\partial T} \right)_V ~ dT $$ Also, ##\left( \dfrac{\partial U}{\partial T} \right)_V = C_V## is specific heat at constant volume.

Therefore, ## 0 = C_V ~ dT + \left( \dfrac{\partial U}{\partial V} \right)_T ~ dT ##.

Using the "First internal energy equation" given above, I calculated that (for Dieterici EoS, as given in the question) ## \left( \dfrac{\partial U}{\partial V} \right)_T = T \left( \dfrac{\partial P}{\partial T} \right)_V - P = \dfrac{Pa}{RTV}##.

[I have double verified my calculation, but still I request to verify that again.]

Using given equation of state, ## P = \dfrac{nRT}{V} e^{-\frac{a}{RTV}} ##. So, ## \left( \dfrac{\partial U}{\partial V} \right)_T = \dfrac{na}{V^2} e^{-\frac{a}{RTV}} ##

Therefore, I obtain the expression ## 0 = C_V ~ dT + \left( \dfrac{na}{V^2} e^{-\frac{a}{RTV}} \right) dV ##. Writing it in differential equation form,

$$ \dfrac{dV}{dT} = - \dfrac{C_V V^2}{na} e^{\frac{a}{RTV}} $$.

Now, I am not sure how to solve this differential equation, where I have to take limits for volume ##\frac{V}{2}## to ##V## and temperature from ##T_0## to temperature to be obtained ##T##.

Problem with the obtained differential equation is that, I am not getting it in a seperable form. Also, I know very little differential equation theory which could be helpful in solving this.I also tried Wolfram Mathemtica DSolve function, but it is also getting very large expressions.

The question is given here https://snboseplc.com/thermodynamics-tifr/ - Question number

**22.**The answer given here is ## T-\left(\frac{na}{C_V}\right)\ln{2} ##. How do I obtain this answer ? Is there any mistake in my approach, or there can be other approach ?Even if the answer is not accurate, how do I obtain the simplified answer ? How do people manage to get temperature change in Engineering thermodynamics to deal with such problems?Any help would be appreciable. Thanks.

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