The initial state of a quantity of monatomic ideal gas is P=1 atm, V=1 liter, and T=373 K. The gas is isothermally expanded to a volume of 2 liters and is then cooled at constant pressure to the volume V. This volume is such that a reversible adiabatic compression to a pressure of 1 atm returns the system to its initial state. All of the changes of state are conducted reversibly. Calculate the value of V and the total work done on or by the gas.
Answers
Answer:
V is approximately 1.52 liters
The work done on the gas = 37 J
Explanation:
The given information are;
Type of gas = Monoatomic gas
p₁ = 1 atm = 101325 Pa
v₁ = 1 liter = 0.001 m³
T₁ = 373 K
v₂ = 2 liters = 0.002 m³
Final volume = V
For isothermal expansion, we have, Boyle's law given as follows;
p₁×v₁ = p₂×v₂
∴ p₂ = p₁×v₁/(v₂)
p₂ = 1 atm × (1 liter)/(2 liters) = 0.5 atm = 50,662.5 Pa
We have for adiabatic compression, we have;
At V, P = p₂ = 0.5 atm (The gas is cooled at constant pressure) and can be reversed back adiabatically to p₁, v₁
Therefore we have;
\dfrac{p_1}{p_2} = \left [\dfrac{V}{v_1} \right ]^\gamma
p
2
p
1
=[
v
1
V
]
γ
γ = 1.66 for a monoatomic gas, which gives;
\dfrac{1 \ atm}{0.5 \ atm} = \left [\dfrac{V}{1 \ liter} \right ]^{1.66}
0.5 atm
1 atm
=[
1 liter
V
]
1.66
V = 1 \ liter \times \sqrt[1.66]{\dfrac{1 \ atm}{0.5 \ atm}} = 1 \ liter \times \sqrt[1.66]{2} \approx 1.52 \ litersV=1 liter×
1.66
0.5 atm
1 atm
=1 liter×
1.66
2
≈1.52 liters
V ≈ 1.52 liters = 0.00152 m³
The total work done is given given by the following relation;
W = \dfrac{K \times \left ( v_f^{1-\gamma}-v_i^{1-\gamma} \right )}{1 - \gamma}W=
1−γ
K×(v
f
1−γ
−v
i
1−γ
)
K = p \times v^{\gamma } = 0.5 \times \sqrt[1.66]{2} ^{1.66 } = 50,662.5 \times (0.00152)^{1.66} \approx 1.06\ Pa \cdot m^{4.98}K=p×v
γ
=0.5×
1.66
2
1.66
=50,662.5×(0.00152)
1.66
≈1.06 Pa⋅m
4.98
W = \dfrac{1.06\times \left ( 0.00152^{{1-1.66}} -0.001^{1-1.66} \right )}{1 - 1.66} \approx 36.9798 \approx 37 \ JW=
1−1.66
1.06×(0.00152
1−1.66
−0.001
1−1.66
)
≈36.9798≈37 J
Given that the work done is positive, we have that work is done in the gas
Explanation:
Step 1-Light Dependent. CO2 and H2O enter the leaf.
Step 2- Light Dependent. Light hits the pigment in the membrane of a thylakoid, splitting the H2O into O2.
Step 3- Light Dependent. The electrons move down to enzymes.
Step 4-Light Dependent. ...
Step 5-Light independent. ...
Step 6-Light independent. ...
calvin cycle.