Isochoric Process with Unknown Ideal Gas

[Preston]

I was assigned a problem in my Engineering Thermodynamics class as follows:
Problem:
An ideal gas in a rigid closed container undergoes isochoric heating from T₁ = 27 C to T₂ = 77 C. Initial gage pressure is 300 kPa, pressure of surroundings is 1 atm.
Find the final gage pressure.

The way to go about solving the problem seems straight forward, as I will show my train of thought below. The only thing I am stuck up on is what R value to use for this particular ideal gas, since the gas itself is not specified, nor it's volume, nor it's specific volume, nor it's mass.

Relevant Equations:
Pv = RT

Attempted Solution:

Here's how I attempted it:

Knowns: T₁ = 27 C, T₂ = 77 C, P_gage1 = 300 kPa, P_surround = 1 atm
Trying to find: P_gage2

For an ideal gas, Pv = RT, where v = specific volume (m³/kg) and R = specific gas constant (J/kgK). If I know what R value to use, I can solve for v:

v = (RT₁)/P₁ where P₁ = P_gage1+P_surround (in Pa)

Then, because volume V is constant, and the unknown arbitrary mass m is constant, so is v
by the relationship

v = V/m

Then I would find P₂:

P₂ = (RT₂)/v

It follows,

P_gage2 = P₂ - P_surround (in Pa)

That's all folks. Maybe I am missing something rather obvious here, or taking the wrong approach. Again, the only thing I am stuck on is what the specific gas constant R should be. I can not solve for it with it's definition since R = R/M = nR where R is the universal gas constant (8.314 J/molK), M is molar mass of the gas, and n is moles of the gas. I have consulted Tables in the back of my text, and R is given for monatomic or diatomic ideal gas as is to be expected, but again, I'm not given those specifics in the problem. Any help is appreciated! :)

- Preston

 

[Bystander]

Find the final gage pressure.

If I know what R value to use, I can solve for v:

Why?

 

[Preston]

Why solve for v, or why am I justified in saying that?

v = (RT₁)/P₁ and everything on right hand side is known (if I know what R to use). Why am solving for v in the first place--

I can use the fact that v is constant through the process (remember, the system is a constant volume closed container) to solve for P₂ (absolute, not gage)

P₂ = (RT₂)/v again, everything on the right is known at this point. Knowing the final absolute pressure allows me to solve for final gage pressure

P_gage2 = P₂ - P_surround

The last two equations were in my original post, but I hope this clarifies my thought process for you.

 

[Bystander]

v = (RT1)/P1

P2 = (RT2)/v

(remember, the system is a constant volume closed container)

Think.

 

[Chestermiller]

True or false: The values of n, R, and V are the same in the initial and final states of the gas.

Chet

 

[Preston]

Chet, that is true. Bystander, I see what you are getting at now!

In both cases, (Pv)/RT = 1

so (P1v)/(RT1) = P2v/RT2 and R and v will cancel out of each side showing that the following relationship is independent of them:

P1/T1 = P2/T2 and now I can solve. Thats what I get for still being up at 3 am when I woke up 6:30 am. Ha.

Thanks so much!