Constant pressure specific heats when temperature changes

[Imolopa]

Homework Statement

Im trying to understand what would be the correct approach for calculating the constant pressure specific heat for an ideal gas undergoing a process where the temperature is changing.

The reason I am asking is because the equation used to calculate C_p0 is dependent on the temperature. So when you want to calculate the work or/and heat exchange for a process where the temperature is changing over time, would you still use this equation and in that case what temperatures would you input? Maybe sum up the start temperature with the end temperature and divide the result by 2? That last method doesn't seem like it would be a very accurate way of doing it, therefore I believe there must some other way that is commonly used?

Homework Equations

C_p0 = C₀ + C₁θ + C₂θ² + C₃θ³

where θ = T /1000 and C_i = constants (gas dependent) found in tables[/B]

 

[Chestermiller]

Homework Statement

Im trying to understand what would be the correct approach for calculating the constant pressure specific heat for an ideal gas undergoing a process where the temperature is changing.

The reason I am asking is because the equation used to calculate C_p0 is dependent on the temperature. So when you want to calculate the work or/and heat exchange for a process where the temperature is changing over time, would you still use this equation and in that case what temperatures would you input? Maybe sum up the start temperature with the end temperature and divide the result by 2? That last method doesn't seem like it would be a very accurate way of doing it, therefore I believe there must some other way that is commonly used?

Homework Equations

C_p0 = C₀ + C₁θ + C₂θ² + C₃θ³

where θ = T /1000 and C_i = constants (gas dependent) found in tables[/B]

You integrate to get the enthalpy change.

 

[Imolopa]

You integrate to get the enthalpy change.

Yeah so with enthalpy values found in tables for the start and end temperature would basically give us what we need in other words the equation: ₁Q₂ = m(h₂-h₁)

Combining with the relation:

₁Q₂= m(u₂-u₁ ) + ₁W₂

where ₁W₂ = mC_v(T₂-₁) = mP(v₂-v₁)

So solving for C_v in short.

 

[Chestermiller]

Yeah so with enthalpy values found in tables for the start and end temperature would basically give us what we need in other words the equation: ₁Q₂ = m(h₂-h₁)

Yes.

Combining with the relation:

₁Q₂= m(u₂-u₁ ) + ₁W₂

where ₁W₂ = mC_v(T₂-₁) = mP(v₂-v₁)

This last equation is incorrect.

 

[Imolopa]

ah yeah the correct ones would be, thank you!:
₁Q₂ = m(u₂-u₁ ) + ₁W₂ = mC_v(T₂-T₁) + ₁W₂

where ₁W₂ = mP(v₂-v₁)

 

[Chestermiller]

ah yeah the correct ones would be, thank you!:
₁Q₂ = m(u₂-u₁ ) + ₁W₂ = mC_v(T₂-T₁) + ₁W₂

where ₁W₂ = mP(v₂-v₁)

Are you trying to determine Cv? The equation you gave certainly doesn't determine $\Delta U$ correctly.

 

[Imolopa]

Are you trying to determine Cv? The equation you gave certainly doesn't determine $\Delta U$ correctly.

Yes that is right I want to determine Cv.

 

[Chestermiller]

Yes that is right I want to determine Cv.

Well, $C_v(\theta)=C_v(\theta)-R$, where R is the universal gas constant and these are the molar heat capacities..

 

[Imolopa]

Well, $C_v(\theta)=C_v(\theta)-R$, where R is the universal gas constant and these are the molar heat capacities..

Ok so conclusionwise either choose a temperature and use in combination with above formula to find Cp which is inputted in the last one to get Cv or calculate using enthalpy with h2 and h1.

 

[Chestermiller]

Ok so conclusionwise either choose a temperature and use in combination with above formula to find Cp which is inputted in the last one to get Cv or calculate using enthalpy with h2 and h1.

I have no idea what you're saying.

 

[Imolopa]

Let's get back to my initial question, and now let's assume that we don't have the values for enthalpy h available.

Given the following relation that is approximately true:
h₂ - h₁ = Cp(T2-T1)

Now given that C_p is a function of the temperature T by the formula:
C_p0 = C₀ + C₁θ + C₂θ² + C₃θ³
where θ = T_i /1000 and C_j = constants (gas dependent) found in tablesSo with the above in mind my question is if it is valid, and more accurate to actually rewrite the top formula to become:
h₂ - h₁ = Cp(T₂) T₂ - Cp(T₁) T₁
In short words use the corresponding C_p for each temperature, rather than just the same C_p for the different temperatures. If not how can one find a C_p that is accurate enough taking into account the different temperatures in the process?

 

[Chestermiller]

Your proposed equation is not accurate unless Cp is constant. Even if Cp varies linearly with temperature, it will give the wrong answer. Here are two more accurate versions that are both exact if Cp varies linearly with temperature:
$$\Delta h=\frac{C_p(T_1)+C_p(T_2)}{2}(T_2-T_1)$$
$$\Delta h=C_p|_{\frac{(T_1+T_2)}{2}}(T_2-T_1)$$

 

[Imolopa]

Thank you!
So to conclude we can say that it is more accurate to use enthalpies instead of constant specific heats, right?

 

[Chestermiller]

Thank you!
So to conclude we can say that it is more accurate to use enthalpies instead of constant specific heats, right?

Sure, unless you have an equation for the average specific heat as a function of the two end point temperatures.