An ideal monatomic gas initially has a temperature of 338 K and a pressure of 6.96 atm. It is to expand from volume 435 cm3 to volume 1310 cm3. If the expansion is isothermal, what are (a) the final pressure (in atm) and (b) the work done by the gas? If, instead, the expansion is adiabatic, what are (c) the final pressure (in atm) and (d) the work done by the gas?
To solve this problem, we need to use the ideal gas law, along with the formulas for isothermal and adiabatic expansions.
The ideal gas law is given by:
PV = nRT
Where:
- P is the pressure of the gas
- V is the volume of the gas
- n is the number of moles of the gas
- R is the ideal gas constant
- T is the temperature of the gas in Kelvin
(a) To find the final pressure during isothermal expansion, we can use the ideal gas law and set the initial and final volumes equal to each other:
P_initial * V_initial = P_final * V_final
We are given the initial pressure (P_initial = 6.96 atm) and volume (V_initial = 435 cm^3). We also know that the expansion is isothermal, meaning the temperature remains constant. Therefore, the final temperature (T_final) is the same as the initial temperature (T_initial = 338 K). We are asked to find the final pressure (P_final).
Substituting the given values into the equation, we can solve for P_final:
P_final = (P_initial * V_initial) / V_final
(b) The work done by a gas during isothermal expansion is given by the equation:
W = -nRT * ln(V_final / V_initial)
Where ln denotes the natural logarithm. We can substitute the known values to find the work done by the gas.
(c) For adiabatic expansion, we need to use a different equation. The adiabatic expansion equation relates the pressure and volume changes:
P_initial * V_initial^γ = P_final * V_final^γ
Where γ is the heat capacity ratio, which depends on the specific gas being used. For a monatomic ideal gas, γ is equal to 5/3.
We are given the initial pressure (P_initial = 6.96 atm) and volume (V_initial = 435 cm^3). We also know the final volume (V_final = 1310 cm^3). We need to find the final pressure (P_final).
Substituting the given values into the equation and solving for P_final, we can find the answer.
(d) To calculate the work done by the gas during adiabatic expansion, we can use the following equation:
W = (P_final * V_final - P_initial * V_initial) / (1 - γ)
Where γ is the heat capacity ratio (5/3 for monatomic gases). We can substitute the known values to find the work done by the gas.