Question

Calculate the work, in joules, done by a gas as it expands at constant temperature from a volume of \(3.00 \mathrm{~L}\) and a pressure of 5.00 atm to a volume of 8.00 L. The external pressure against which the gas expands is \(1.00 \mathrm{~atm} .(1 \mathrm{~atm}=101,325 \mathrm{~Pa})\)

Short answer

-506.625 Joules

Step-by-step solution

Converting the given pressure to Pascals

Firstly, convert the external pressure against which the gas expands from atmospheres to Pascals using the conversion factor given: 1 atm = 101,325 Pa. Thus, the external pressure is 1 atm x 101,325 Pa/atm = 101,325 Pa.

Converting the initial and final volumes to cubic meters

Then, convert the given volumes from liters to cubic meters. Since 1 L = 0.001 m³, the initial volume (Vi) is 3.00 L x 0.001 m³/L = 0.003 m³ and the final volume (Vf) is 8.00 L x 0.001 m³/L = 0.008 m³.

Calculating the work done by the gas

Calculate the work done by the gas using the formula W = -Pext x ΔV, where Pext is the external pressure (in Pascals) and ΔV is the change in volume (in cubic meters). The work done is negative because the gas is doing work on the surroundings. Therefore, W = -101,325 Pa x (0.008 m³ - 0.003 m³) = -101,325 Pa x 0.005 m³ = -506.625 J.

Key concepts

Constant Temperature Expansion

During constant temperature expansion, a gas expands, changing its volume while maintaining a steady temperature. This process is commonly referred to as an isothermal expansion. To visualize how this works, imagine a balloon being slowly inflated; the pressure within the balloon adjusts as it expands to ensure its temperature remains constant.

Under these conditions, the gas's internal energy does not change because the added heat energy precisely compensates for the work done by the gas. By adhering to one of the gas laws, which we’ll discuss later, the relationship between pressure and volume during an isothermal process for an ideal gas is described by Boyle's law. Boyle’s law states that the pressure of a given mass of gas is inversely proportional to its volume when temperature is constant. For your textbook problem, you'd use the equation that expresses the work done by a gas under these conditions, incorporating constant temperature as a critical aspect of solving the problem.

Pressure-Volume Work

Pressure-volume work, or P-V work, is the work performed by or on a gas when it undergoes a volume change under an external pressure. It’s a central concept in thermodynamics when examining how energy is transferred between a system and its surroundings.

The amount of work done is calculated using the formula: \( W = -P_{\text{ext}} \times \Delta V \), where \( W \) is the work, \( P_{\text{ext}} \) is the external pressure, and \( \Delta V \) is the change in volume. The minus sign indicates that work done by the system on its surroundings is considered negative within the conventions of thermodynamics. In the context of your textbook exercise, this formula helps determine the energy transferred as the gas expanded against a constant atmospheric pressure.

Gas Laws

Gas laws are mathematical relationships that describe the behavior of gases and how they respond to changes in temperature, volume, and pressure. They are essential for understanding various physical processes, such as those involving the expansion or compression of gases.

There are several foundational gas laws, including Boyle's Law, Charles's Law, and the combined gas law, that can be further summarized by the ideal gas law equation \( PV=nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the amount of gas in moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. These laws assume the gas behaves 'ideally,' meaning its molecules are small compared to the space between them and there are no interactions between the molecules apart from collisions.

Unit Conversion

Unit conversion is the process of changing the expression of a measurement to a different unit without altering the actual amount. In physics and engineering, it's essential to work with consistent units to apply formulas correctly and interpret results accurately.

For example, in your textbook problem, you needed to convert atmospheric pressure from 'atm' to 'Pascals' and volume from 'liters' to 'cubic meters' to use the standard SI units. The ability to convert between units is pivotal, not just for solving textbook problems but across all scientific disciplines to ensure clear communication and accurate calculations.