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!
