Question

In: Physics

Im trying to derive Cp - Cv = R where Cp and Cv are the molar...

Im trying to derive Cp - Cv = R where Cp and Cv are the molar specific heats at constant pressure and volume respectively. (R = 8.314 J/(mol*K)). So I'm only having problems with the last step.

I get that for a process at constant pressure we have:

∆E(internal) = Q + W = nCp∆T - P∆V

And I get that for a process at constant volume we have:

∆E(internal) = Q + W = Q + 0 => ∆E(internal) = Q

The last step is setting these two equations equal to one another, but I don't understand how we know that the internal energy of both processes are the same.

Solutions

Expert Solution

For an Ideal gas the internal energy is function of temperature only. This is very important.

Let us consider one mole of an ideal gas enclosed in a cylinder provided with a frictionless piston of area A. Let P, V and T be the pressure, volume and absolute temperature of gas respectively as shown in below figure.

A quantity of heat dQ is supplied to the gas. To keep the volume of the gas constant, a small weight is placed over the piston. The pressure and the temperature of the gas increase to P + dP and T + dT respectively. This heat energy dQ is used to increase the internal energy dE of the gas. But the gas does not do any work (dW = 0).

∴ dQ = dE = 1 × Cv × dT …... (1)

The additional weight is now removed from the piston. The piston now moves upwards through a distance dx, such that the pressure of the enclosed gas is equal to the atmospheric pressure P. The temperature of the gas decreases due to the expansion of the gas.

Now a quantity of heat dQ’ is supplied to the gas till its temperature becomes T + dT. This heat energy is not only used to increase the internal energy dE of the gas but also to do external work dW in moving the piston upwards.

∴ dQ’ = dE + dW

Since the expansion takes place at constant pressure,

dQ ′ = CpdT

∴ CpdT = CvdT + dW …... (2)

Work done, dW = force × distance = = P × A × dx

dW = P dV (since A × dx = dV, change in volume)

∴ CpdT = CvdT + P dV …... (3)

The equation of state of an ideal gas is ( for 1 mole )

PV = RT

Differentiating both the sides

PdV = RdT …... (4)

Substituting equation (4) in (3),

CpdT = CvdT + RdT

Cp - Cv = R

For any query regarding the answer, let me know.

genius_generous answered 1 hour ago
Next > < Previous

Related Solutions

Coding in R Im trying to plot a histogram in R, and i know the basic...
Coding in R Im trying to plot a histogram in R, and i know the basic code for that, however, the code requires 2 plots coming from the same data set. for example, the voltage of an old process vs. a new process. is there any ways to seperate the 2 processes out from the data set to make 2 relative frequencyt histograms?
A specific type of ideal gas has a specific heat capacity at constant pressure (cp=cv+R) that...
A specific type of ideal gas has a specific heat capacity at constant pressure (cp=cv+R) that is a function of temperature T, such that cp=0.48T+885, where cp has units of J/kg/K and T has units of K. The gas, which is initially at T1 = 314 K and P1 = 1x105 Pa, undergoes a reversible adiabatic process such that its final temperature is T2 = 772 K. Calculate the pressure of the gas (in Pa) in this final state. Assume...
Im trying to create a function in C where it takes an array and size and...
Im trying to create a function in C where it takes an array and size and returns the largest absolute value in the array (either negative or positive) using only stdio.h. Any help would be greatly appreciated! Ex: if the array is { -2.5, -10.1, 5.2, 7.0}, it should return -10.1 Ex: if the array is {5.1, 2.3, 4.9, 1.0}, it should return 5.1. double getMaxAbsolute(double array[], int size) {     double max = array[0];    double abs[size];    for...
Im trying to get a GUI interface in java where there is four text fields for...
Im trying to get a GUI interface in java where there is four text fields for the first name, last name, department, and phone number of employee in a program. Then also have a radio button below for Gender (Male/Female/Other) and a list for the Title (Mr./Ms./Mrs./Dr./Col./Prof.). At the very bottom of the frame there has to be buttons for printing, submitting and exiting but for whatever reason when I tried it nothing appears in the frame regardless of what...
Show that Cp = T(∂S/∂T)p and Cv = T(∂S/∂T)V
Show that Cp = T(∂S/∂T)p and Cv = T(∂S/∂T)V
A 2.00 mol sample of an ideal gas with a molar specific heat of CV =...
A 2.00 mol sample of an ideal gas with a molar specific heat of CV = 5 2 R always starts at pressure 1.00 ✕ 105 Pa and temperature 350 K. For each of the following processes, determine the final pressure (Pf, in kPa), the final volume (Vf, in L), the final temperature (Tf, in K), the change in internal energy of the gas (ΔEint, in J), the energy added to the gas by heat (Q, in J), and the...
A 2.00 mol sample of an ideal gas with a molar specific heat of CV =...
A 2.00 mol sample of an ideal gas with a molar specific heat of CV = (5/2)R always starts at pressure 2.00 ✕ 105 Pa and temperature 300 K. For each of the following processes, determine the final pressure (Pf, in kPa), the final volume (Vf, in L), the final temperature (Tf, in K), the change in internal energy of the gas (ΔEint, in J), the energy added to the gas by heat (Q, in J), and the work done...
A 1.50 mol sample of an ideal gas with a molar specific heat of CV =...
A 1.50 mol sample of an ideal gas with a molar specific heat of CV = 5/2 R always starts at pressure 2.00 ✕ 105 Pa and temperature 250 K. For each of the following processes, determine the final pressure (Pf, in kPa), the final volume (Vf, in L), the final temperature (Tf, in K), the change in internal energy of the gas (ΔEint, in J), the energy added to the gas by heat (Q, in J), and the work...
A 2.50 mol sample of an ideal gas with a molar specific heat of CV =...
A 2.50 mol sample of an ideal gas with a molar specific heat of CV = 5/2 R always starts at pressure 1.50 • 10^5 Pa and temperature 300 K. For each of the following processes, determine the final pressure (Pf, in kPa), the final volume (Vf, in L), the final temperature (Tf, in K), the change in internal energy of the gas (ΔEint, in J), the energy added to the gas by heat (Q, in J), and the work...
a) Why do the symbols for heat capacity have suffixes? b) CV is different from CP....
a) Why do the symbols for heat capacity have suffixes? b) CV is different from CP. Which is larger and why? c) Why is the difference between CV and CP often neglected for solids?
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT