A small sphere of initial volume V is filled with n moles of helium at initial pressure and temperature P and T.

Complete Question:

A small sphere of initial volume V is filled with n moles of helium at initial pressure and temperature and T. Which of the following statements is true?

a) The volume decreases to V/2, and the pressure increases to 4P when the temperature is T/2

b) n/2 moles of gas are removed, the volume is decreased to V/2, and the pressure decreases to P/4 with a drop in temperature of T/2

c) n moles of gas are added, the total sample is heated to 2T, and the pressure drops to P/2 when the volume increases to 8V

d) The amount of gas is doubled to 2n, the pressure is doubled to 2P, and the volume is doubled to 2V, with a corresponding temperature drop to T/2

Answer:

c

Explanation:

Let's consider the helium as an ideal gas, so it can be studied by the ideal gas law, which states:

PV = nRT

Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant and T is the temperature. Because R is constant:

PV/nT = R. Thus the initial state must be equal to the final state.

So, let's check the statements:

a) Let's indicate the final state as P₂, V₂, n₂ and T₂. So, if T₂ = T/2:

PV/nT = P₂V₂/n₂T₂

PV/nT = P₂V₂/n₂ (T/2)

PV/nT = 2P₂V₂/n₂T

So, if V₂ = V/2 and P₂ = 4P:

PV/nT = 2 * (V/2 * 4P) / n2T

PV/nT = 4VP/n2T

Which is not correct!

b) Now, if T₂ = T/2:

PV/nT = 2P₂V₂/n₂T

If n/2 is removes, n₂ = n/2. And, V₂ = V/2 and P₂ = P/4:

PV/nT = 2 * (V/2 * P/4) / (n/2) * T

PV/nT = 4 * (V/2 * P/4) / nT

PV/nT = PV/2nT

Which is not corret!

c) Now, if V₂ = 8V:

PV/nT = P₂*8V/n₂T₂

And n₂ = n + n = 2n, T₂ = 2T and P₂ = P/2:

PV/nT = (P/2) * 8V/2n*2T

PV/nT = 8 * (PV) / 2*2n*2T

PV/nT = 8 * (PV) / 8 * (nT)

PV/nT = PV/nT

So, it's correct!

d) Now, T₂ = T/2, n₂ = 2n, P₂ = 2P, and V₂ = 2V:

PV/nT = 2P*2V/2n * (T/2)

PV/nT = 4PV/nT

Which is not correct!