Gaseous hydrogen contained initially under standard conditions in a sealed vessel of volume V = 5.01 L was cooled by `DeltaT=50 K`. Find how much the internal energy of the gas will change and what amount of heat will be lost by the gas.

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the change in internal energy (ΔU)
The change in internal energy for an ideal gas can be calculated using the formula:
\[ \Delta U = n C_V \Delta T \]
where:
- \( n \) = number of moles of the gas
- \( C_V \) = molar specific heat at constant volume
- \( \Delta T \) = change in temperature

### Step 2: Determine the number of moles (n)
Using the ideal gas law:
\[ PV = nRT \]
We can rearrange this to find \( n \):
\[ n = \frac{PV}{RT} \]
Given:
- \( P = 1 \, \text{atm} = 1.01 \times 10^5 \, \text{Pa} \)
- \( V = 5.01 \, \text{L} = 5.01 \times 10^{-3} \, \text{m}^3 \)
- \( R = 8.31 \, \text{J/(mol K)} \)
- \( T = 298 \, \text{K} \)

Substituting the values:
\[ n = \frac{(1.01 \times 10^5 \, \text{Pa})(5.01 \times 10^{-3} \, \text{m}^3)}{(8.31 \, \text{J/(mol K)})(298 \, \text{K})} \]

### Step 3: Calculate the number of moles (n)
Calculating \( n \):
\[ n = \frac{(1.01 \times 10^5)(5.01 \times 10^{-3})}{(8.31)(298)} \approx 0.0204 \, \text{mol} \]

### Step 4: Determine the molar specific heat at constant volume (C_V)
For diatomic gases like hydrogen, the degrees of freedom \( f = 5 \). Therefore,
\[ C_V = \frac{f}{2} R = \frac{5}{2} R = \frac{5}{2} \times 8.31 \approx 20.78 \, \text{J/(mol K)} \]

### Step 5: Calculate the change in internal energy (ΔU)
Now we can substitute \( n \), \( C_V \), and \( \Delta T \) into the equation for ΔU:
\[ \Delta U = n C_V \Delta T \]
Given \( \Delta T = -50 \, \text{K} \):
\[ \Delta U = (0.0204 \, \text{mol})(20.78 \, \text{J/(mol K)})(-50 \, \text{K}) \]
Calculating:
\[ \Delta U \approx 0.0204 \times 20.78 \times -50 \approx -21.2 \, \text{J} \]

### Step 6: Calculate the heat lost (ΔQ)
Since the work done \( W = 0 \) (sealed vessel), we can use the first law of thermodynamics:
\[ \Delta U = \Delta Q + W \]
Thus,
\[ \Delta Q = \Delta U \]
Therefore, the heat lost by the gas is:
\[ \Delta Q \approx -21.2 \, \text{J} \]

### Final Answer
The change in internal energy of the gas is approximately \(-21.2 \, \text{J}\) and the amount of heat lost by the gas is also approximately \(-21.2 \, \text{J}\).

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