5 moles of ideal gas cloud has an initial pressure of 1.00 bar, initial volume of 100.0L and temperature of 25.0ºC. The cloud expands adiabatically to a final volume of 900.0L. Cp,m= 15.79 J mol-1 K-1 .
What is the final pressure and final temperature of the gas cloud?
What is the change in entropy for this process?
It would be really helpful if you could explain how do I get started on this problem and what formula i supposed to use Thanks.
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To solve this problem, we can use the equations of adiabatic processes and the ideal gas law. Let's start by finding the final pressure and temperature of the gas cloud.
1. First, we need to find the initial and final temperatures in Kelvin. The initial temperature is given as 25.0ºC, so we need to convert it to Kelvin by using the formula:
T(K) = T(ºC) + 273.15
T(initial) = 25.0 + 273.15 = 298.15 K
2. To find the final temperature, we will use the adiabatic process equation:
PV^γ = constant
where P is the pressure, V is the volume, and γ is the heat capacity ratio. For an ideal monoatomic gas, γ is equal to 5/3.
Now, we can use the ideal gas law equation to find the final pressure:
P(initial) * V(initial)^γ = P(final) * V(final)^γ
Substituting the given values:
1.00 * (100.0)^5/3 = P(final) * (900.0)^5/3
Solving for P(final):
P(final) = 1.00 * (100.0)^5/3 / (900.0)^5/3
3. Now that we have the final pressure, we can find the final temperature using the ideal gas law equation:
PV = nRT
where n is the number of moles and R is the ideal gas constant (8.314 J/mol·K).
Rearranging the equation to solve for T(final):
T(final) = P(final) * V(final) / nR
Substituting the given values:
T(final) = P(final) * V(final) / (5 * 8.314)
Now, let's calculate the final pressure and temperature.
Final pressure (P(final)):
Substitute the given values into the equation:
P(final) = 1.00 * (100.0)^5/3 / (900.0)^5/3
Calculate the result.
Final temperature (T(final)):
Substitute the given values into the equation:
T(final) = P(final) * V(final) / (5 * 8.314)
Calculate the result.
Finally, let's calculate the change in entropy.
The change in entropy (ΔS) for an adiabatic process can be calculated using the equation:
ΔS = Cp * ln(T(final)/T(initial))
where Cp is the molar heat capacity at constant pressure.
Substitute the given values into the equation and calculate the result.
Remember to check if any unit conversions are needed, especially for the gas constant value.
I hope this explanation helps you understand how to solve the problem.