One mole of an ideal gas(Cv=3/2R) is expanded adiabatically until the temperature drops from 20°C to 10°C. Calculate ∆H for the process.

To find the change in enthalpy (∆H) for the adiabatic expansion of one mole of an ideal gas, we can use the relationship between enthalpy, temperature, and the specific heat at constant volume (Cv). In this case, you've provided that Cv = (3/2)R, where R is the universal gas constant. Let's break down the calculation step by step.

Understanding the Basics

For an ideal gas, the change in enthalpy (∆H) can be expressed as:

∆H = n * Cp * ∆T

where:

• n = number of moles of the gas (in this case, n = 1 mole)
• Cp = specific heat at constant pressure
• ∆T = change in temperature (in Kelvin)

Finding Cp

For an ideal gas, the relationship between Cp and Cv is given by:

Cp = Cv + R

Substituting the value of Cv:

Cp = (3/2)R + R = (5/2)R

Calculating the Temperature Change

Next, we need to convert the temperature change from Celsius to Kelvin. The initial temperature (T1) is 20°C, and the final temperature (T2) is 10°C:

• T1 = 20 + 273.15 = 293.15 K
• T2 = 10 + 273.15 = 283.15 K

The change in temperature (∆T) is:

∆T = T2 - T1 = 283.15 K - 293.15 K = -10 K

Calculating ∆H

Now we can substitute the values into the enthalpy change equation:

∆H = n * Cp * ∆T

Substituting the known values:

∆H = 1 * (5/2)R * (-10 K)

∆H = -25R

Final Result

To express the final result, we can use the value of R, which is approximately 8.314 J/(mol·K):

∆H = -25 * 8.314 J/(mol·K) = -207.85 J

Thus, the change in enthalpy (∆H) for the adiabatic expansion of one mole of the ideal gas, as the temperature drops from 20°C to 10°C, is approximately -207.85 J. This negative value indicates that the process is exothermic, meaning heat is released during the expansion.