Problem 82 - A Complete Resource Book in Chemistry for JEE Main

Question

To calculate the amount of work done in joules during a reversible isothermal expansion of an ideal gas, the volume must be expressed in (a) \(\mathrm{dm}^{3}\) only (b) \(\mathrm{m}^{3}\) only (c) \(\mathrm{cm}^{3}\) only (d) any one of them

Step-by-Step Solution

Answer

(d) any one of them

Step 1: Understanding the Work Formula in Isothermal Expansion

For an isothermal (constant temperature) expansion of an ideal gas, the work done is calculated using the formula:\[ W = -nRT \ln\left(\frac{V_f}{V_i}\right) \]where \(W\) is the work done, \(n\) is the number of moles, \(R\) is the universal gas constant, \(T\) is the absolute temperature, \(V_f\) is the final volume, and \(V_i\) is the initial volume. Notice that the volumes appear in a ratio, \(\frac{V_f}{V_i}\), which is dimensionless.

Step 2: Analyzing Volume Units

Since the ratio \(\frac{V_f}{V_i}\) is dimensionless, the units of \(V_f\) and \(V_i\) cancel each other out. This means any consistent unit of volume can be used, as they cancel out when taking the ratio. Thus, when calculating work, we can use any unit of volume (\(\mathrm{dm}^3\), \(\mathrm{m}^3\), or \(\mathrm{cm}^3\)) as long as both \(V_f\) and \(V_i\) are in the same unit.

Step 3: Universal Gas Constant Considerations

When performing calculations with the gas constant \(R\), ensure consistency in units. For example, if using \(R = 8.314\, \mathrm{J \cdot mol^{-1} \cdot K^{-1}}\), where the energy is in joules, volume should ideally be in cubic meters (\(\mathrm{m}^3\)), as Joules inherently align with SI units. However, since the volume units cancel in the ratio, conversion factors between \(\mathrm{dm}^3\), \(\mathrm{m}^3\), or \(\mathrm{cm}^3\) won't affect the ratio, so the primary concern is maintaining consistency.

Step 4: Final Conclusion

The volume in the work formula can be measured in any consistent units provided the initial and final volumes \(V_i\) and \(V_f\) use the same units, due to their ratio being dimensionless.

Key Concepts

Ideal Gas

An ideal gas is a theoretical concept used in physics and chemistry to simplify the study of gases. It's based on the ideal gas law, which describes the relationship between pressure (\( P \)), volume (\( V \)), temperature (\( T \)), and the number of moles (\( n \)) of a gas. This relationship is given by the equation:\[ PV = nRT \]In this formula, the ideal gas behaves according to certain assumptions:

• The gas molecules do not interact with each other except during elastic collisions.
• The volume of gas molecules is negligible compared to the space between them.

These assumptions allow us to predict and calculate the behavior of gases under a variety of conditions, making the concept of an ideal gas very useful in many scientific and engineering applications. However, it's important to note that real gases deviate from ideal behavior under high pressures and low temperatures.

Work Done

In thermodynamics, work done by or on a system is a measure of energy transfer. During an isothermal expansion of an ideal gas, work is done by the system. The formula to calculate the work done (\( W \)) during such a process is:\[ W = -nRT \ln\left(\frac{V_f}{V_i}\right) \]This equation tells us:

• \( n \) is the number of moles of the gas.
• \( R \) is the universal gas constant.
• \( T \) is the absolute temperature (constant in isothermal processes).
• \( V_f \) and \( V_i \) are the final and initial volumes, respectively.

The negative sign indicates that work is done by the gas during the expansion. This means the system is losing energy. Understanding this concept is crucial in applications like engines and refrigeration systems.

Volume Units

Volume can be measured in several units like cubic meters (\( m^3 \)), cubic decimeters (\( dm^3 \)), or cubic centimeters (\( cm^3 \)). In the formula for work done during isothermal expansion, the term \( \frac{V_f}{V_i} \) is dimensionless. This means:

• The specific units of volume don't matter as long as \( V_f \) and \( V_i \) are in the same units.
• This feature is particularly useful because it allows us flexibility depending on the context of the problem.

For consistency and to avoid confusion, it's often recommended to use the same unit systems throughout calculations. In many scientific contexts, using \( m^3 \) is preferred because it aligns with the International System of Units (SI). Nevertheless, the choice depends on what makes the calculations more straightforward or aligns with other constants being used.

Universal Gas Constant

The universal gas constant (\( R \)) is a fundamental part of the ideal gas law. It serves as the bridge between the macroscopic and molecular levels of gases. The constant value depends on the units used in calculations. Some common values are:

• \( 8.314 \, \mathrm{J \, mol^{-1} \, K^{-1}} \)
• \( 0.0821 \, \mathrm{L \, atm \, mol^{-1} \, K^{-1}} \)

When applying the ideal gas law or calculating work during isothermal changes, it's crucial to ensure that all other units in the equation are consistent with the chosen value of \( R \). For instance, if you are using \( R = 8.314 \, \mathrm{J \, mol^{-1} \, K^{-1}} \), you'll need to ensure that your volumes are expressed in \( m^3 \), since a joule is based on the metric system. This consistency allows accurate calculations and prevents errors in engineering and scientific computations.