How Do PV Diagrams Illustrate the Behavior of Ideal Gases?

[JQ10]

Homework Statement

1 litre of a monatomic ideal gas starts from an initial state (i) at T = 300K and P = 10^5 N/m^2. It (the gas) is changed to a final state (f), such as that its pressure decreases 20% and its volume double.

a) Draw a PV Diagram mark its initial state and final state upon it


b) Calculate the temperature of its final state


c) Calculate the change in internal energy caused by transition from the initial state to the final state.

d) Calculate the work done during the transition.

Homework Equations

Ideal Gas Laws
N = n*Na
(3/2kT) * number of molecules

The Attempt at a Solution

a) Find initial volume by using V = 1/P so 10^-5 m^3 then to find the final volume find double this so 1/50000 m^3 and find the final pressure by finding 4/5 of 10^5 which is 80000. Then go ahead plotting these point on the graph I have drawn with relevant scales.

b) Find the number of molecules in the gas with N = n*Na (6.02*10^-23) so after using n = PV/kT (10^5 * 10^5 / 1.38*10^-23 * 300 = 2.5*10^20) to get n. This gives 0.015 (2.5*10^20 * 6.02*10^-23)

Then I did Tf = (PfVf)/kN ((80000*1/50000)/1.38*10^-23 * 0.015) to get 7.73*10^25K (I'm sure this is wrong)

c) Use (3/2kT) * number of molecules but I don't know whether to use the initial or final temperature.

d) Use W = integrate sign v2 v1 PdV (do you use the initial or final pressure?)



Help would be appreciated very much.

 

[JaWiB]

a) Find initial volume by using V = 1/P so 10^-5 m^3 then to find the final volume find double this so 1/50000 m^3 and find the final pressure by finding 4/5 of 10^5 which is 80000. Then go ahead plotting these point on the graph I have drawn with relevant scales.

The problem says the volume is initially 1L, which is .001 m^3

b) Find the number of molecules in the gas with N = n*Na (6.02*10^-23) so after using n = PV/kT (10^5 * 10^5 / 1.38*10^-23 * 300 = 2.5*10^20) to get n. This gives 0.015 (2.5*10^20 * 6.02*10^-23)

n = PV / RT (R is the gas constant)

c) Use (3/2kT) * number of molecules but I don't know whether to use the initial or final temperature.

It asks for the change so you're looking for U_final - U_initial

 

[kerimek]

First of all, it is important to note that the ideal gas law only applies to gases at low pressure and high temperature, so it may not be accurate for this situation. However, for the purpose of this exercise, we will assume that it is applicable.

a) Your approach to drawing the PV diagram and marking the initial and final states is correct.

b) To find the final temperature, we can use the ideal gas law, PV = nRT. Rearranging this equation to solve for T, we get T = PV/nR. Plugging in the values for the final state (Pf = 80,000 N/m^2, Vf = 2 m^3, n = 0.015 mol, R = 8.314 J/mol*K), we get Tf = 6,000 K.

c) The change in internal energy can be calculated using the formula ΔU = (3/2)nRT. Plugging in the values for the final state (n = 0.015 mol, R = 8.314 J/mol*K, Tf = 6,000 K), we get ΔU = 449,100 J.

d) To calculate the work done during the transition, we can use the formula W = ∫PdV. Since the process is a change in pressure and volume, we can use the trapezoidal rule to approximate the integral. Plugging in the values for the initial and final states (P1 = 100,000 N/m^2, P2 = 80,000 N/m^2, V1 = 10^-5 m^3, V2 = 2*10^-5 m^3), we get W = -1,500 J.

I hope this helps clarify any confusion and helps you in your calculations. Remember to always double check your units and make sure they are consistent throughout the problem. Good luck with your studies!