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?
(a) In an isothermal expansion,
P*V = constant = Po*Vo
P(final)/Po = Vo/V(final)
= 435/1310 = 0.332
Use that and the initial pressure to get the final pressure.
(b) Work done by gas =
Integral of P*dV = (Po*Vo)*Integral dV/V
= Po*Vo ln(1310/435)
Make sure P is in units of Pascals, and Volume in m^3, to get work in Joules.
(c) For adiabatic expansion of a monatomic gas,
P*V^(5/3) = constant = Po*Vo^(5/3)
Pfinal/Po = (0.332)^5/3 = 0.159
(d) Calculate the P*dV integral again, this time using the adiabatic relationship for P(V).
P (V) = [Po*Vo^(5/3)]/V^5/3
To solve this problem, we can use the ideal gas law and the formulas for work done in isothermal and adiabatic processes.
(a) Final Pressure in the isothermal expansion:
In an isothermal process, the temperature remains constant. So, we can use the formula:
P1 * V1 = P2 * V2
Where:
P1 = Initial Pressure = 6.96 atm
V1 = Initial Volume = 435 cm^3
V2 = Final Volume = 1310 cm^3
Rearranging the formula, we have:
P2 = (P1 * V1) / V2
Substituting the values, we get:
P2 = (6.96 atm * 435 cm^3) / 1310 cm^3
P2 ≈ 2.32 atm
The final pressure in the isothermal expansion is approximately 2.32 atm.
(b) Work done by the gas in the isothermal expansion:
In an isothermal process, the work done by the gas can be calculated using the formula:
Work = P * ΔV * ln(V2 / V1)
Where:
P = Pressure = 6.96 atm (Initial pressure)
ΔV = Change in Volume = V2 - V1
V2 = Final Volume = 1310 cm^3
V1 = Initial Volume = 435 cm^3
Substituting the values, we get:
Work = 6.96 atm * (1310 cm^3 - 435 cm^3) * ln(1310 cm^3 / 435 cm^3)
Work ≈ 14469.18 atm * cm^3
The work done by the gas in the isothermal expansion is approximately 14469.18 atm * cm^3.
(c) Final Pressure in the adiabatic expansion:
In an adiabatic process, the relationship between pressure and volume can be expressed as:
P1 * V1^γ = P2 * V2^γ
Where:
P1 = Initial Pressure = 6.96 atm
V1 = Initial Volume = 435 cm^3
P2 = Final Pressure (to be determined)
V2 = Final Volume = 1310 cm^3
γ = Adiabatic index (specific heat ratio) for monatomic gas ≈ 5/3
Rearranging the formula, we have:
P2 = (P1 * V1^γ) / V2^γ
Substituting the values, we get:
P2 = (6.96 atm * (435 cm^3)^(5/3)) / (1310 cm^3)^(5/3)
P2 ≈ 4.30 atm
The final pressure in the adiabatic expansion is approximately 4.30 atm.
(d) Work done by the gas in the adiabatic expansion:
In an adiabatic process, the work done by the gas can be calculated using the formula:
Work = (P1 * V1 - P2 * V2) / (γ - 1)
Where:
P1 = Initial Pressure = 6.96 atm
V1 = Initial Volume = 435 cm^3
P2 = Final Pressure = 4.30 atm
V2 = Final Volume = 1310 cm^3
γ = Adiabatic index (specific heat ratio) for monatomic gas ≈ 5/3
Substituting the values, we get:
Work = (6.96 atm * 435 cm^3 - 4.30 atm * 1310 cm^3) / (5/3 - 1)
Work ≈ -3292.57 atm * cm^3
The work done by the gas in the adiabatic expansion is approximately -3292.57 atm * cm^3. (Note that the negative sign indicates that work is done on the gas rather than being done by the gas.)
To solve this problem, we can use the ideal gas law equation, which is given by:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature.
(a) To find the final pressure in an isothermal expansion, we can use the equation:
P1 * V1 = P2 * V2
where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume, respectively.
Given:
P1 = 6.96 atm
V1 = 435 cm^3
V2 = 1310 cm^3
Using the equation above, we can rearrange it to solve for P2:
P2 = (P1 * V1) / V2
Substituting the known values, we have:
P2 = (6.96 atm * 435 cm^3) / 1310 cm^3
P2 = 2.32 atm
So, the final pressure is 2.32 atm.
(b) To find the work done by the gas in an isothermal expansion, we can use the equation:
Work = nRT * ln(V2/V1)
Since the expansion is isothermal, the temperature remains constant. Therefore, we can simplify the equation to:
Work = nRT * ln(V2/V1)
To calculate the work done by the gas, we need to know the number of moles of gas, n. However, this information is not provided in the problem statement. Without the value of n, we cannot calculate the work done in this scenario.
(c) To find the final pressure in an adiabatic expansion, we can use the equation:
P1 * V1^γ = P2 * V2^γ
where γ is the adiabatic index, which depends on the specific gas. For a monatomic ideal gas, γ is equal to 5/3.
Given:
P1 = 6.96 atm
V1 = 435 cm^3
V2 = 1310 cm^3
γ = 5/3
Using the equation above, we can rearrange it to solve for P2:
P2 = (P1 * V1^γ) / V2^γ
Substituting the known values, we have:
P2 = (6.96 atm * (435 cm^3)^(5/3)) / (1310 cm^3)^(5/3)
P2 ≈ 1.92 atm
So, the final pressure in an adiabatic expansion is approximately 1.92 atm.
(d) To find the work done by the gas in an adiabatic expansion, we can use the equation:
Work = (P2 * V2 - P1 * V1) / (γ - 1)
Given:
P1 = 6.96 atm
V1 = 435 cm^3
P2 = 1.92 atm
V2 = 1310 cm^3
γ = 5/3
Using the equation above, we can substitute the known values to calculate the work done by the gas:
Work = (1.92 atm * 1310 cm^3 - 6.96 atm * 435 cm^3) / (5/3 - 1)
Work ≈ 3195.6 cm^3·atm
So, the work done by the gas in an adiabatic expansion is approximately 3195.6 cm^3·atm.