An ideal monatomic gas initially at 298 K and 1 atm pressure is
expanded from a volume of 1 liter to 2 liters. Consider separately
isobaric, isothermal and adiabatic expansions to accomplish this. In each
case, calculate the following:
a) The final temperature of the gas
b) The amount of work done by the gas in Joules
c) The amount of heat added to the gas in Joules
d) The change in energy of the gas in Joules
I've tried to do this multiple times but I'm really confused about the equations and when to apply them.
To solve this problem, you need to apply the appropriate equations based on the type of expansion (isobaric, isothermal, and adiabatic) and use the gas laws.
1. Isobaric Expansion:
a) The final temperature of the gas can be found using the formula:
Final temperature (T2) = Initial temperature (T1) * (Final volume (V2) / Initial volume (V1))
In this case, T2 = 298 K * (2 L / 1 L) = 596 K
b) The work done by the gas can be calculated using the formula:
Work done (W) = Pressure (P) * Change in volume (ΔV)
Since the pressure is constant (isobaric), W = P * (V2 - V1) = 1 atm * (2 L - 1 L) = 1 atm * 1 L = 101.3 J
c) The amount of heat added to the gas can be calculated using the formula:
Heat added (Q) = n * R * Change in temperature (ΔT)
Where n is the number of moles of gas and R is the universal gas constant.
Since the process is isobaric, the ΔT can be found as:
ΔT = Final temperature (T2) - Initial temperature (T1) = 596 K - 298 K = 298 K
We need the number of moles (n) to calculate Q.
d) The change in energy of the gas is equal to the sum of work done and the amount of heat added:
Change in energy (ΔE) = W + Q
2. Isothermal Expansion:
a) The final temperature of the gas remains constant, so it's the same as the initial temperature of 298 K.
b) The work done by the gas can be calculated using the formula (since it's an ideal gas):
W = n * R * Temperature * ln(V2 / V1)
W = n * 8.314 J/mol·K * 298 K * ln(2 L / 1 L) = n * 8.314 J/mol·K * 298 K * ln(2)
c) The amount of heat added to the gas is zero because the temperature remains constant.
d) The change in energy is equal to the work done:
ΔE = W
3. Adiabatic Expansion:
a) The final temperature of the gas during an adiabatic process can be calculated using the formula:
T2 = T1 * (V1 / V2)^((γ - 1) / γ)
Where γ is the heat capacity ratio and depends on the specific gas. For a monatomic ideal gas, γ = 5/3.
b) The work done by the gas can be calculated using the formula:
W = (P1 * V1 - P2 * V2) / (γ - 1)
In an adiabatic process, the pressure decreases as the volume increases, so P1 * V1 > P2 * V2.
c) The amount of heat added to the gas is zero because the process is adiabatic.
d) The change in energy is equal to the work done:
ΔE = W
Remember to convert the units as necessary.
I hope this explanation helps you understand the equations and how to apply them for each type of expansion.