a gas diatomic gas is heated at constant pressure.the percentage of the total heat supllied for work is

When a diatomic gas is heated at constant pressure, it undergoes a process where part of the heat supplied goes into doing work, while the rest increases the internal energy of the gas. To understand how to calculate the percentage of heat supplied that is used for work, we need to delve into some thermodynamic principles.

Understanding the Basics

In thermodynamics, when a gas is heated at constant pressure, the heat added to the system can be expressed using the formula:

• Q = ΔU + W

Where:

• Q is the total heat supplied.
• ΔU is the change in internal energy.
• W is the work done by the gas.

Work Done by the Gas

For a gas expanding at constant pressure, the work done can be calculated using the formula:

• W = PΔV

Here, P is the pressure and ΔV is the change in volume. Since we are dealing with a diatomic gas, we can also relate the change in internal energy to the temperature change:

• ΔU = nC_vΔT

Where:

• n is the number of moles of the gas.
• C_v is the molar heat capacity at constant volume.
• ΔT is the change in temperature.

Calculating the Percentage of Heat Supplied for Work

For a diatomic gas, the molar heat capacities are:

• C_v = (5/2)R
• C_p = (7/2)R

Using the relationship between heat capacities, we can express the total heat supplied at constant pressure as:

• Q = nC_pΔT = n(7/2)RΔT

Now, substituting the expressions for ΔU and W into the first equation:

• Q = nC_vΔT + PΔV

Since at constant pressure, PΔV = nRΔT, we can rewrite the equation as:

• Q = n(5/2)RΔT + nRΔT

This simplifies to:

• Q = n(7/2)RΔT

Finding the Work Percentage

Now, to find the percentage of heat supplied that is used for work:

• Percentage for Work = (W/Q) × 100

Substituting the values:

• W = nRΔT
• Q = n(7/2)RΔT

Thus, the percentage becomes:

• Percentage for Work = (nRΔT / n(7/2)RΔT) × 100 = (2/7) × 100

This results in:

• Percentage for Work ≈ 28.57%

Final Thoughts

In summary, when a diatomic gas is heated at constant pressure, approximately 28.57% of the total heat supplied is used for work. Understanding these relationships helps in grasping the broader concepts of thermodynamics and the behavior of gases under different conditions.