One mole of a gas enclosed in a vessel is heated at constant pressure through 1K. Work done by the gas is

(a) 1J

(b) 1/R J

(c) R J

Please explain the solution in detail.

To determine the work done by one mole of a gas when it is heated at constant pressure through a temperature change of 1 K, we can use the principles of thermodynamics. The key concept here is the relationship between pressure, volume, and temperature for an ideal gas. Let's break this down step by step.

Understanding the Basics

When a gas is heated at constant pressure, it expands. The work done by the gas during this expansion can be calculated using the formula:

• W = PΔV

Where:

• W = work done by the gas
• P = pressure of the gas
• ΔV = change in volume

Relating Volume Change to Temperature Change

For an ideal gas, we can relate the change in volume to the change in temperature using the ideal gas law:

• PV = nRT

Where:

• n = number of moles of gas
• R = universal gas constant (approximately 8.314 J/(mol·K))
• T = absolute temperature in Kelvin

Since we have one mole of gas (n = 1), the equation simplifies to:

• PV = RT

Now, if we increase the temperature by 1 K (ΔT = 1 K), we can find the change in volume (ΔV) at constant pressure. From the ideal gas law, we can derive:

• ΔV = nRΔT / P

Calculating Work Done

Substituting ΔV back into the work formula gives us:

• W = P * (nRΔT / P)

Notice that the pressure (P) cancels out:

• W = nRΔT

Since we have one mole of gas and the temperature change is 1 K, we can substitute:

• W = 1 * R * 1

This simplifies to:

• W = R

Final Answer

Therefore, the work done by the gas when it is heated at constant pressure through a temperature change of 1 K is equal to R J. Given the options provided, the correct answer is (c) R J.

Summary

In summary, when one mole of an ideal gas is heated at constant pressure, the work done is directly proportional to the temperature change and the universal gas constant. This relationship highlights the fundamental principles of thermodynamics and the behavior of gases under varying conditions.