Solving a Student's Experiment on an Ideal Gas: Molar Specific Heat Capacity

[Maori_Rights]

Homework Statement

A student performs an experiment on an ideal gas by adding 42.0 J of heat to it. As a result the student finds that the volume of the gas changes from 50 cm3 to 150 cm3 while the pressure remains constant at 101.3 kPa.

i) If the quantity of the gas present is 0.007 moles, determine the molar specfic heat capacity of the gas that the student would find at constant pressure.

ii) What is the molar specific heat capacity of the gas at constant volume?

2 Relevant equations

For Gases:

PV = nRT

Q = n . Cv . ΔT for constant-volume processes
Q = n . Cp . ΔT for constant-pressure processes

Cp = Cv + R

ΔU = Q - W

W = P . ΔV for an isobaric process

Q = 42J
Vi = 50 cm3 = 50 μm3
Vf = 150 cm3 = 150μm3
P = 101300 Pa
n = .007 moles
R = 8.315 J / (mol K)

The Attempt at a Solution

First, I worked out the work done on the system by the surroundings:

Since this is an isobaric expansion,

W = P . ΔV
= 101300 * (150 - 50)μ
= 10.13 J

so ΔU is;

ΔU = Q - W
= 42 - 10.13
= 31.87 J.

I am not sure why I calculated the change in internal energy. Do I even use it in this whole process?


"Q (constant pressure) must account for both the increase in internal energy and the transfer of energy out of the system by work"

for (i) the molar specific heat capacity at a constant pressure,

Q = n . Cp . ΔT

so Cp = Q / ( n . ΔT )


and PV = nRT

so T = ( PV ) / ( nR)

=> ΔT = T_f - T_i
= { ( PV_f ) / ( nR) } - { ( PV_i ) / ( nR) }
= ( P / nR ) ( V_f - V_i)
= ( 101300 / (0.007 * 8.315) ) ( (150-50)μ)
= 174.0400 K

So C_p is;

C_p = Q / ( n . ΔT )
= 42 / ( 0.007 * 174.0400)
= 34.47482725 J mol^-1 K ^-1

Therefore the molar specific heat capacity of the gas at a constant pressure of 101300 Pa is 34.5 J mol^-1 K ^-1

I think that that might be the right answer, but is my reasoning correct? I'm not too sure since I have been spending an age trying to work this one out and I keep getting different values, I have approached several "classmates" who turn their backs at the call for help, but typing slowy and going through this whole process of listing everyting down, I think I may have finally cracked it. Can you please verify this.
*For all the PHY160s out there who can't find a willing helper, this one is for you. I sincerely do hope that all of you, even those who would prefer not to share methods, get that 2% from this online assesment which would be so ever useful in your bid to get invited to the Medical Interviews*



As for (ii) What is the molar specific heat capacity of the gas at constant volume?


I am not sure how I can solve this using:

Q = n . Cv . ΔT for constant-volume processes

However, after patiently churning through my notes, admist all the equations and explanations in monochrome, I found the formula and note:

Cp = Cv + R "NB: This is true for ALL conditions."

So C_v is;

C_v = C_p - R
= 34.47482725 J mol^-1 K ^-1 - 8.315 J mol^-1 K ^-1
= 26.15982725
= 26.2 J mol^-1 K ^-1

Thefore the molar specific heat capacity of the gas at constant volume is 26.2 J.

This answer also agrees with the provided answer. I know that the note says that "this is true for all conditions" so does this mean that it is true for, even, this condition?

And is there any other way to solve a problem like this?


Thanks for all your help. Even writing this post has helped me alot!

 

[TSny]

The OP's solutions for both parts appear to me to be correct.

There is no need to calculate the temperature change. For a constant pressure process of an ideal gas, the ideal gas law yields $n \Delta T =\frac { P \Delta V}{R}$. So, $$C_p = \frac{Q}{n \Delta T} = \frac{Q R}{P \Delta V}$$