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 during the expansion is (c) \(-954.2 \, J\).

Step-by-step solution

Understand the Process

This is an isothermal process, meaning the temperature remains constant. Therefore, the work done by an ideal gas during an isothermal expansion can be calculated using the formula: \[ W = -nRT \ln\left(\frac{V_f}{V_i}\right) \]where \( n \) is the number of moles, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, \( V_f \) is the final volume, and \( V_i \) is the initial volume.

Convert Temperature to Kelvin

Conversion from Celsius to Kelvin is necessary for thermodynamic calculations. The temperature \( T \) in Kelvin is given by \[ T = 27 + 273.15 = 300.15 \, K \]

Apply Values to the Formula

Now use the formula for work done during isothermal expansion:- Number of moles \( n = 0.75 \)- Gas constant \( R = 8.314 \, J \, K^{-1} \, mol^{-1} \)- Temperature \( T = 300.15 \, K \)- Initial volume \( V_i = 15 \, L \)- Final volume \( V_f = 25 \, L \)Plug these into the formula:\[ W = - 0.75 \times 8.314 \times 300.15 \times \ln \left(\frac{25}{15}\right) \]

Calculate the Natural Logarithm

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

Calculate the Work Done

Substitute the values from Step 3 and the logarithm from Step 4 into the formula and compute:\[ W = - 0.75 \times 8.314 \times 300.15 \times 0.5108 \approx -954.2 \, J \]

Identify the Correct Option

Compare the calculated work done to the given options. The correct value is option (c) \(-954.2 \, J\).

Key concepts

Ideal Gas

An ideal gas is a theoretical concept where a gas follows certain rules perfectly. These rules, known as the Ideal Gas Law, combine several measurable properties like pressure, volume, and temperature, into a straightforward equation.
This equation is expressed as \[ PV = nRT \] where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles of the gas
• \( R \) is the gas constant
• \( T \) is the temperature

In an ideal gas, the molecules are assumed to not interact with each other and occupy no volume themselves. This simplification allows us to easily predict how a gas will behave under different conditions.

Isothermal Expansion

Isothermal processes occur when the temperature of a system remains constant. An isothermal expansion is specifically where a gas increases in volume without a change in temperature.
During isothermal expansion, the gas needs to do work to push against external pressure, which often requires heat to flow into the system to maintain temperature equilibrium. The work done by the gas can be calculated using the formula:\[ W = -nRT \, \ln \left( \frac{V_f}{V_i} \right) \] In this equation:

• \( V_f \) is the final volume
• \( V_i \) is the initial volume
• \( \ln \) is the natural logarithm function

This formula reflects that more work is done if the gas expands significantly, as the volume ratio gets larger.

Work Done by Gas

Work done by a gas during expansion or compression is a key concept in thermodynamics. When a gas expands, it can do work on its surroundings by pushing against them.
For an isothermal expansion, the work done is negative, indicating that the energy is leaving the gas as it does work on the surroundings. The formula \[ W = -nRT \ln \left( \frac{V_f}{V_i} \right) \] allows us to determine this work. The negative sign reflects that the gas is losing energy in doing work.
Absolute units of work done are measured in joules. In processes where the gas returns to its initial state, like a full cycle of expansion and compression, the net work done can be calculated from area under a pressure-volume curve.

Gas Constant

The gas constant, often symbolized as \( R \), is a fundamental constant in physics and chemistry. It bridges the macroscopic properties of gases (those we measure like pressure and volume) with their molecular properties (like energy levels).
It is crucial in calculations involving ideal gases, as seen in the ideal gas law and during processes like isothermal expansion.

• In our exercise, \( R \) is provided as \( 8.314 \, \text{J} \, \text{K}^{-1} \, \text{mol}^{-1} \).
• This value helps convert energy terms into consistent units of measurement across moles, temperature, and volume.

Understanding the role of \( R \) lets us predict how a gas will react to changes in temperature, pressure, and volume, making it an indispensable tool in chemical and physical analysis.