Ideal Gas Exercise: Homework Statement & Solution

[Perrin]

Homework Statement

A cylinder with a frictionless piston is placed horizontally in an atmosphere pressure 1 * 10⁵ N/m². A gas in the cylinder is initially at a temperature of 300K with a volume of 6.0 * 10^-3 m³. Then the gas is heated slowly to 400K. How much work is done by the gas in the process?

Homework Equations

Ideal Gas equations:

p*V/T = constant

p*V = NKT = nRT

Ek (average) = (3/2) * kT

Ek (total) = (3/2) * NkT = (3/2)pV

The Attempt at a Solution

At first I calculated the initial energy as:
E₁ = (3/2)pV = (3/2)*(1 * 10⁵) * (6.0 * 10^-3) = 900J

Then, assuming the pressure is constant, I said:

pV₁/T₁ = pV₂/T₂

V₂ = V₁*T₂ / T₁

V₂ = 6.0 * 10^-3 * 400 / 300 = 8*10^-3

Thus, the second energy:

E₂ = (3/2)*10⁵*8*10^-3 = 1200J

And the work:
W = E₂ - E₁ = 1200 - 900 = 300J.

Now, in the answer to the question, it says the work is not 300J, but 200J.
Can someone enlighten me about my mistake?

Thanks.

 

[Ygggdrasil]

You calculated the change in internal energy of the gas (ΔE). ΔE = w + q, so you are ignoring q, the amount of heat transferred during the process. You should use this equation for work instead:

$$w = -\int_{v_1}^{v_2}{P_{ext}dV}$$

 

[Perrin]

So, I could also calculate:

w = ΔE - q ?

And could you please explain to me how to use the equation you wrote? What's Pext? How do you integrate it? Sorry, I'm not very good with integrals...

Thanks for the quick reply!

 

[Ygggdrasil]

P_ext is the external pressure. If the external pressure stays constant during your process, P does not vary with V and you can pull it out of the integral to get the simpler relation w = -P_extΔV. If P varies as the volume changes, you have to do the integration by finding an expression for P in terms of V.

 

[Perrin]

Thanks, I finally understand :D