Number of moles necessary to get piston back to initial position

In summary, the conversation discusses the relationships between pressure, volume, and temperature in a gas system. Part (a) calculates the temperature and work done by the gas on the spring, while part (b) calculates the change in entropy of the gas. Part (c) uses the calculated values to determine the final temperature and number of moles of gas. Overall, the conversation demonstrates the principles of thermodynamics in a gas system.
  • #1
lorenz0
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Homework Statement
In a cylindrical piston of section S a piston can move without friction. A mole of ideal monoatomic gas is confined to the left, the right chamber is empty and contains a spring with elastic constant k which would be at rest if the piston were all the way to the left.
Initially the gas is at rest in an "A" state (##p_A = 10^5 Pa,
V_A = 25 dm^3##). It takes place through an exchange of heat with the outside
a slow transformation until reaching a "B" state with volume
halved. Calculate :
a) the heat ##Q## exchanged by the gas with the external environment to complete the transformation,
b) the entropy change ##\Delta S_G## of the gas,
c) how many moles ##n_{agg}## of gas must be added to bring the piston back to the initial position assuming that the temperature remains that of state "B"?
Relevant Equations
##F_{spring}=-kx, \Delta U=Q-L, W=\int_{a}^{b} \vec{F}\cdot\vec{dx}, \Delta S=nC_V\ln(\frac{T_f}{T_i})+nR\ln(\frac{V_f}{V_i}), PV=nRT##
a) ##T_A=\frac{p_AV_A}{nR}=300.7K, P_A V_A=kL^2=nRT_A##, ##P_B S=k\frac{L}{2}\Rightarrow P_B V_B=k(\frac{L}{2})^2 \Rightarrow P_B=\frac{kL^2}{2V_A}=\frac{P_AV_A}{2V_A}=\frac{P_A}{2}##, ##W_{spring\to gas}=\int_{L}^{L/2}kxdx=-\frac{3}{8}kL^2=-\frac{3}{8}nRT_A####\Rightarrow Q=L+\Delta U=-\frac{3}{8}nRT_A+n\cdot\frac{3}{2}R (\frac{T_A}{4}-T_A)\simeq 3750 J##

b) ##\Delta S_{gas}=nC_V\ln(\frac{T_f}{T_i})+nR\ln(\frac{V_f}{V_i})=nC_V\ln(\frac{T_A /4}{T_A})+nR\ln(\frac{V_A/2}{V_A})=-4nR\ln(2)=-23 J/K##

EDIT: I think I have managed to solve also part c)

c) ##T_B=\frac{P_BV_B}{nR}=\frac{1}{nR}\cdot\frac{kL^2}{2V_A}\cdot\frac{V_A}{2}=\frac{1}{4}\frac{kL^2}{nR}=\frac{1}{4}\cdot\frac{P_AV_A}{nR}=\frac{1}{4}T_A## so ##n_f=\frac{P_AV_A}{RT_B}=\frac{10^5\cdot 25\cdot 10^{-3}}{8.314\cdot\frac{300.7}{4}}mol=4mol## so ##\Delta n=n_f-n_i=(4-1)mol=3mol##
 

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  • #2
In part (a), your W is the work done by the gas on the springbecause ##Q=W+\Delta U##. So, for Q, I get $$Q=-1.5RT_A=-3750\ J$$

The rest looks good. Very nice job.
 
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Related to Number of moles necessary to get piston back to initial position

1. What is the equation for calculating the number of moles necessary to get a piston back to its initial position?

The equation is: n = P(V1-V2)/RT, where n is the number of moles, P is the pressure, V1 is the initial volume, V2 is the final volume, R is the gas constant, and T is the temperature.

2. How do you determine the pressure and volume values for the equation?

The pressure and volume values can be determined experimentally by measuring the pressure and volume of the gas in the piston before and after it is moved back to its initial position.

3. What is the significance of the gas constant in the equation?

The gas constant, R, is a proportionality constant that relates the pressure, volume, and temperature of a gas. It allows us to convert between different units of measurement and is necessary for accurately calculating the number of moles.

4. Can the equation be used for any type of gas?

Yes, the equation can be used for any type of gas as long as the units for pressure, volume, and temperature are consistent (e.g. all in SI units or all in imperial units).

5. Is the number of moles necessary to get the piston back to its initial position dependent on the size of the piston?

No, the number of moles necessary is not dependent on the size of the piston. It is dependent on the pressure, volume, and temperature of the gas inside the piston.

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