Question

If \(0.75\) mole of an ideal gas is expanded isothermally at \(27^{\circ} \mathrm{C}\) from 15 litres to 25 litres, then work done by the gas during this process is \(\left(\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right)\) (a) \(-1054.2 \mathrm{~J}\) (b) \(-896.4 \mathrm{~J}\) (c) \(-954.2 \mathrm{~J}\) (d) \(-1254.3 \mathrm{~J}\)

Short answer

The work done by the gas is \(-954.2 \text{ J}\), matching option (c).

Step-by-step solution

Understand the Isothermal Expansion Equation

In isothermal expansion, the temperature of the system remains constant. The work done by an ideal gas during an isothermal expansion can be calculated using the formula for isothermal processes:\[ W = nRT \ln \left(\frac{V_f}{V_i}\right) \]where \( W \) is the work done, \( n \) is the number of moles, \( R \) is the universal gas constant, \( V_f \) is the final volume, and \( V_i \) is the initial volume.

Substitute Known Values into Equation

Given that \( n = 0.75 \) moles, \( V_i = 15 \text{ L} = 0.015 \text{ m}^3 \) and \( V_f = 25 \text{ L} = 0.025 \text{ m}^3 \). The temperature is \( 27^{\circ} \text{C} \), which needs to be converted into Kelvin: \( T = 27 + 273 = 300 \text{ K} \). Using the formula:\[ R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1} \]Substitute the known values into the equation:\[ W = 0.75 \times 8.314 \times 300 \times \ln \left(\frac{0.025}{0.015}\right) \]

Solve the Natural Logarithm Component

Calculate the natural logarithm part first:\[ \ln \left(\frac{0.025}{0.015}\right) = \ln \left(\frac{25}{15}\right) = \ln \left(\frac{5}{3}\right) \approx 0.5108 \]

Calculate the Work Done

Now, compute the work done using the previously obtained values:\[ W = 0.75 \times 8.314 \times 300 \times 0.5108 \]\[ W \approx 0.75 \times 8.314 \times 300 \times 0.5108 \approx 954.2 \text{ J} \]

Determine the Sign of the Work Done

In an isothermal expansion, the work done by the system is considered to be negative because the gas is doing work on the surroundings. Thus, the value of \( W \) is \(-954.2 \text{ J} \).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the relationship between pressure, volume, temperature, and the amount of gas. It is expressed as \( PV = nRT \), where \( P \) denotes pressure, \( V \) stands for volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) represents temperature in Kelvin.
For an isothermal process, which is a process that occurs at a constant temperature, the relationship remains governed by this law, but the pressure and volume can change as long as their product remains the same, maintaining the condition \( T = ext{constant} \).
This law is critical in understanding the behavior of gases under different conditions, allowing predictions about how changes in one state variable can affect the others. Understanding this relationship helps elucidate how gases expand, such as during the isothermal expansion when the temperature stays the same but the volume increases.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of isothermal expansion, thermodynamics helps explain how energy transfer happens within a system.
Key concepts of thermodynamics include:

• Zeroth Law: If two systems are in thermal equilibrium with a third system, they are in equilibrium with each other.
• First Law: Also known as the law of energy conservation, it states that energy cannot be created or destroyed, only transformed. For an isothermal expansion, the energy transferred as work must be accounted for by heat transfer to the system.
• Second Law: The total entropy, or disorder, of an isolated system cannot decrease over time. This explains why heat flows spontaneously from hot to cold objects.
• Third Law: As temperature approaches absolute zero, the entropy of a perfect crystal reaches a constant minimum.

Thermodynamics provides the logical framework to predict, quantify, and describe physical processes such as the expansion of gases.

Work Done in Thermodynamics

Work done in thermodynamics refers to the energy transfer that occurs when an external force is applied to a system, causing displacement. When dealing with gases, such work is often considered during volume changes.
During an isothermal expansion, work is done by a gas on its surroundings. For ideal gases, this work can be calculated using the formula: \[ W = nRT \, \ln \left(\frac{V_f}{V_i}\right) \] Here:

• \( W \): Work done by the gas
• \( n \): Number of moles of the gas
• \( R \): Universal gas constant \( 8.314 \, \text{J K}^{-1} \text{mol}^{-1} \)
• \( T \): Absolute temperature in Kelvin
• \( V_i \) and \( V_f \): Initial and final volumes, respectively

In isothermal expansion, the temperature remains constant, so the expansion requires the input of heat to perform work, which is why the work done is directly dependent on the natural log of the change in volume ratio. The negative sign for work signifies the gas doing work on the surroundings.