he necessary equations from thermodynamics. Let's break down the steps:

1. Calculate the number of moles of hydrogen gas (H₂) at STP:

At STP (Standard Temperature and Pressure), the volume of one mole of an ideal gas is 22.4 liters. Given the cylinder has a fixed capacity of 44.81 liters, we can find the number of moles, $$n$$:

$$ n = \frac{\text{Volume of gas}}{\text{Volume of one mole at STP}} = \frac{44.81 \, \text{L}}{22.4 \, \text{L/mol}} = 2 \, \text{mol} $$

2. Use the specific heat capacity at constant volume (Cₜ):

For hydrogen gas (H₂), which is diatomic, the molar heat capacity at constant volume $$C_V$$ is:

$$ C_V = \frac{5}{2} R $$

where $$R = 8.31 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1}$$. So:

$$ C_V = \frac{5}{2} \times 8.31 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1} = 20.775 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1} $$

3. Calculate the heat required:

The heat required $$Q$$ to raise the temperature of the gas by $$\Delta T = 20^{\circ} \text{C}$$ (which is the same in Kelvin) is given by the formula:

$$ Q = n \cdot C_V \cdot \Delta T $$

Plugging in the values:

$$ Q = 2 \, \text{mol} \times 20.775 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1} \times 20 \, \text{K} = 831 \, \text{J} $$

Thus, the amount of heat needed to raise the temperature of the hydrogen gas in the cylinder by $$20^{\circ} \text{C}$$ is 831 J.

Therefore, the correct option is:

Option C: 831 J

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