Constant pressure specific heats when temperature changes

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Discussion Overview

The discussion revolves around the calculation of constant pressure specific heats (Cp) for an ideal gas undergoing temperature changes. Participants explore the implications of varying Cp on work and heat exchange calculations, particularly in the context of enthalpy changes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question how to accurately calculate Cp when temperature changes, suggesting methods like averaging the start and end temperatures.
  • Others propose integrating to find enthalpy changes and using enthalpy values from tables for the start and end temperatures.
  • A participant mentions a relation involving enthalpy and internal energy, but another challenges the correctness of the equation presented.
  • There is a discussion about determining Cv and the relationship between Cv and Cp, with references to the universal gas constant R.
  • One participant suggests using different Cp values for different temperatures rather than a single average value, raising questions about the accuracy of this approach.
  • Another participant provides alternative equations for calculating changes in enthalpy when Cp varies with temperature, emphasizing that using a constant Cp may lead to inaccuracies.
  • There is a suggestion that using enthalpy may be more accurate than relying solely on constant specific heats.

Areas of Agreement / Disagreement

Participants express differing views on the best approach to calculate Cp and the validity of using average values versus specific enthalpy values. There is no consensus on a single method, and the discussion remains unresolved regarding the most accurate approach.

Contextual Notes

Participants highlight limitations in their approaches, including the dependence on the assumption of constant Cp and the accuracy of using average values for temperature changes. Some equations presented may not hold under all conditions.

Imolopa
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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 Cp0 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



Cp0 = C0 + C1θ + C2θ2 + C3θ3

where θ = T /1000 and Ci = constants (gas dependent) found in tables[/B]
 
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Imolopa said:

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 Cp0 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



Cp0 = C0 + C1θ + C2θ2 + C3θ3

where θ = T /1000 and Ci = constants (gas dependent) found in tables[/B]
You integrate to get the enthalpy change.
 
Chestermiller said:
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 = mCv(T2-1) = mP(v2-v1)

So solving for Cv in short.
 
Imolopa said:
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 = mCv(T2-1) = mP(v2-v1)
This last equation is incorrect.
 
ah yeah the correct ones would be, thank you!:
1Q2 = m(u2-u1 ) + 1W2 = mCv(T2-T1) + 1W2

where 1W2 = mP(v2-v1)
 
Imolopa said:
ah yeah the correct ones would be, thank you!:
1Q2 = m(u2-u1 ) + 1W2 = mCv(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.
 
Chestermiller said:
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.
 
Imolopa said:
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..
 
Chestermiller said:
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.
 
  • #10
Imolopa said:
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.
 
  • #11
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 Cp is a function of the temperature T by the formula:
Cp0 = C0 + C1θ + C2θ2 + C3θ3
where θ = Ti /1000 and Cj = 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 Cp for each temperature, rather than just the same Cp for the different temperatures. If not how can one find a Cp that is accurate enough taking into account the different temperatures in the process?
 
  • #12
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:
$$\Delta h=\frac{C_p(T_1)+C_p(T_2)}{2}(T_2-T_1)$$
$$\Delta h=C_p|_{\frac{(T_1+T_2)}{2}}(T_2-T_1)$$
 
  • #13
Thank you!
So to conclude we can say that it is more accurate to use enthalpies instead of constant specific heats, right?
 
  • #14
Imolopa said:
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.
 

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