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The Ideal Gas Law is a fundamental equation in thermodynamics that describes the behavior of an ideal gas. This law combines several gas laws to form a single equation: \[ PV = nRT \]where:

• \(P\) is the pressure of the gas, typically measured in pascals (Pa).
• \(V\) is the volume of the gas in cubic meters (m\(^3\)).
• \(n\) is the number of moles of gas.
• \(R\) is the ideal gas constant, approximately \(8.314 \text{ J/(mol} \cdot \text{K)}\).
• \(T\) is the temperature of the gas in Kelvin (K).

This equation relates these variables, making it possible to calculate one if the others are known. For example, in our exercise, it helps us derive how the volume changes with temperature when pressure is held constant.

Work done by a gas during expansion or compression is a crucial concept in thermodynamics. Whenever a gas changes its volume under constant pressure, it performs work. The formula to calculate work done by the gas is:\[ W = P \, \Delta V \]Here, \( \Delta V \) is the change in volume, which can be expressed as \( V_2 - V_1 \).

When the volume increases, the gas does positive work on the surroundings.

Conversely, when the volume decreases, work is done on the gas.

In our specific scenario, given that the volume doubles at constant pressure, the work done can be efficiently determined by calculating \( nRT_1 \) as demonstrated. This simplification arises from substituting the ideal gas conditions into the work formula.

The calculation of moles is an essential step in solving problems involving the ideal gas law. Moles (represented as \( n \)) indicate the amount of substance within the gas. Using the molar mass of a substance allows us to convert between mass and moles:\[ n = \frac{\text{mass}}{\text{molar mass}} \]In our problem:

• The molar mass of hydrogen gas (\( H_2 \)) is approximately \(2.02 \text{ g/mol}\).
• The given mass of hydrogen is \(6.44 \text{ g}\).

Thus, the number of moles is calculated as:\[ n = \frac{6.44 \text{ g}}{2.02 \text{ g/mol}} \approx 3.19 \text{ mol} \]This value is crucial for pinpointing the volume and pressure in further calculations using the ideal gas law.

Constant pressure processes, also known as isobaric processes, play a vital role in thermodynamic studies. These processes occur when the pressure remains unchanged while the other factors—such as volume and temperature—may vary. For gases, the relationship between temperature and volume at constant pressure is governed by Charles's Law:\[ V \propto T \]or, in an equation form: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]This principle aids in understanding how gas expands or contracts under temperature alteration at constant pressure.In the case we discussed, as the temperature of the gas increased, its volume doubled.

• This doubling of volume meant the work done by the gas could be calculated using the ideal gas law's conditions.

Knowing these details allows us to appreciate how the gas performs work when subjected to a rise in temperature capable of maintaining constant pressure through its expansion.