Question

For one mole of an ideal gas, which of these statements must be true?
(a) U and H each depends only on temperature
(b) Compressibility factor z is not equal to 1
(c) \[{C_{p,m}} - {C_{v,m}} = R\]
(d) \[dU = {C_v}dT\] for any process
1) (a), (c) and (d)
2) (a) and (c)
3) (c) and (d)
4) (b), (c) and (d)

Answer 1

Hint: An ideal gas is one in which particles are of negligible volume. The size of the particles is equal and no force of attraction is present between the particles. The particles move at a rapid speed in random directions.

Complete Step by Step Answer:
Let's discuss all the options one by one.
We know that the equation for an ideal gas is PV = nRT, with the increase in temperature, the internal energy of the system increases causing the faster movement of molecules. So, internal energy equates with the heat of the system and depends on the temperature.
And the formula of enthalpy is, \[H = U + PV\] , H stands for enthalpy, U for internal energy, P for pressure and V for volume.
For an ideal gas of 1 mole, we can write PV = RT
So, the equation of enthalpy becomes,
\[H = U + RT\]

Now, the internal energy for an ideal gas is given by C_vΔT.
Hence, the equation for enthalpy thus becomes,
\[H = C_vT + RT\]

From the above equation, we can say that H is dependent only on temperature.
So, statement (a) is true.
For an ideal gas, the compressibility factor (z) is always 1. Therefore, statement (b) is wrong.
For an ideal gas, \[{C_{p,m}} - {C_{v,m}} = R\]. So, statement (c) is true.
The expression of internal energy in terms of heat capacity is written as \[dU = {C_v}dT\] and it is true for all cases.
Therefore, statement (d) is also true.
Therefore, option 1 is true.

Note: It is to be noted that, in an ideal gas, the particles have a perfect elastic collision with no loss of energy. They obey the laws of motion given by Newton. The equation for an ideal gas is PV = nRT, this equation determines whether the gas is ideal or not.