First law of thermodynamics and the ideal gas law

Problem:

5 moles of gas in a cylinder undergo an isobaric expansion starting at 293 K.  The cylinder is initially 50 cm tall.  The radius of the cylinder is 10 cm and Δy is 1 cm.  How much work is done by the gas?

Solution:

Problem:

One mole of an ideal gas does 3000 J of work on its surroundings as it expands isothermally to a final pressure of 1 atm and volume of 25 L.  Determine
(a)  the initial volume and
(b)  the temperature of the gas.

Solution:

Problem:

A cylinder contains He gas.  In its initial state a the cylinder Volume is Va = 4.23*10-3 m3 and the pressure is Pa = 1.19*10-5 Pa.  The volume is isothermically reduced until the system reaches a state b with Vb = 0.581*10-3 m3.  This process is followed by an isobaric expansion which allows the system to reach a state c with Vc = Va.  The cycle finishes with an isochoric or isometric (constant volume) process that brings the system back to state a.
(a)  Draw the cycle in a P-V diagram.
(b)  Find the work done by the system going from b to c.
(c)  Find the work done by the system in one cycle.
(d)  Find the fractional change of the temperature of the system going from b to c.
(e)  When the system goes from c to a, does it absorb or release heat?  Find the amount of heat absorbed or released.
(f)  When the system goes from b to c, does it absorb or release heat?  Find the amount of heat absorbed or released.

Solution:

Problem:

Air at 20.0 oC in the cylinder of a Diesel engine is compressed from an initial pressure of 1.00 atm and volume of 800 cm3 to a volume of 60 cm3.  Assume that air behaves as an ideal gas with γ  = 1.40 and that the compression is adiabatic.  Find the final pressure and temperature of the air.  (Give numerical answers!)

Solution:

Problem:

A quantity of an ideal monatomic gas consist of N atoms, initially at temperature T1
The pressure and volume are then slowly doubled, in such a way as to trace out a straight line on the P-V diagram. 
In terms of N, k, and T1, find
(a)  the work done by the gas.
(b)  If one defines an equivalent specific heat capacity (c = ΔQ/ΔT, where ΔQ is the total heat transferred to the gas) for this particular process for the  above monatomic gas, what is its value?

Solution:

Problem:

Two cylinders A and B, with equal diameters have inside two pistons with negligible mass connected by a rigid rod.
The pistons can move freely.
The rod is a short tube with a valve.  The valve is initially closed.

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Cylinder A and its piston is adiabatically insulated and cylinder B is in thermal contact with a thermostat which has the temperature T = 27o C.
Initially the piston of cylinder A is held fixed and cylinder A contains m = 32 kg of argon at a pressure higher than the atmospheric pressure.  Cylinder B contains oxygen at atmospheric pressure (101.3 kPa).
Releasing the piston of cylinder A, it moves slowly (quasi-statically) until it reaches equilibrium.  At equilibrium the volume of the gas in cylinder A has increased by a factor of eight, and in cylinder B the oxygen's density has increased by a factor of 2.  During this process the heat Q = 747.9*104 J is delivered to the thermostat.

(a)  Use the ideal gas law and the first law of thermodynamics to show that for the process taking place in cylinder A we have TV = constant.
(b)  Calculate the parameters P, V, and T of the argon in the initial and final states.
(c)  After equilibrium has been reached, the valve between the cylinders is opened.   Calculate the final pressure of the mixture of the gases.
The kilo-molar mass of argon is 40 kg/kmol.

Solution:

Problem:

A container with helium is sealed by a movable heavy piston with cross-sectional area A.  Initially, the piston is in equilibrium, the volume of the gas is V, and the system is in thermal equilibrium with its surroundings. The piston is then slowly lifted by an external force a distance L and held until the thermal equilibrium with the surroundings is re-established.  After that, the container is thermally insulated from the surroundings and piston is released.  Find the new equilibrium position of the piston with respect to its initial position.  Will the piston come to rest or will it oscillate about this position?  Neglect friction, the heat capacities of the piston and the container, and the mass of the gas compared to the mass of the piston.

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Solution:

Problem:

One mole of a monatomic gas is heated in such a way that its molar specific heat is 2R.  During the heating, the volume of the gas is doubled.  By what factor does the temperature of the gas change?

Solution:

Problem:

One mole of an ideal gas undergoes transformations that take it from point A to point B on a p-V diagram by two different paths α and β.  Points A and B have the same temperature T = T0 (this means that they lie on an isotherm) and the pressure pA at point A is larger than the pressure pB at point B.  Path α takes the system from state A to state I through an isochoric (constant volume) transformation, followed by an isobaric (constant pressure) transformation from state I to state B.  Path β takes the system from state A to state II through an isobaric transformation, followed by an isochoric transformation from state II to state B.  Assume that pA and pB are known.
(a)  Find VA and VB in terms of the known quantities pA, pB and T0.
(b)  In a p-V diagram plot path α and path β
(c)  Assuming that the heat capacities CV and CP are constants what is the heat flow into the gas for each path?  Give your results in terms of pA, pB, T0, CV and Cp.  In each case is heat absorbed or released by the system?  Why?
(d)  In absolute value, for what process is the heat larger?  Why?

Solution:

Problem:

A horizontal insulated cylinder is partitioned by a frictionless insulated piston which prevents heat exchange between the two sides.  On each side of the piston we have 30 liters of an ideal monatomic gas at a pressure of 1 atmosphere and a temperature of 300 K.  Heat is very slowly injected into the gas on the left, causing the piston to move until the gas on the right is compressed to a pressure of 2 atmosphere.
(a)  What are the final volume and temperature on the right side?
(b)  What are the final values of P, V, and T on the left side?
(c)  What is the change in the internal energy for the gas on the right side?  How much heat did it absorb and how much work was done on it?
(b)  What is the change in the internal energy for the gas on the left side?  How much heat did it absorb and how much work was done on it?

Solution:

Problem:

Let CP and CV denote the molar specific heat of an ideal gas at constant pressure and at constant volume, respectively.  Show that CP - CV = R.

Problem:

Let CP and CV denote the molar specific heat of an ideal diatomic gas at room temperature at constant pressure and constant volume, respectively.  Find the ratio CP/CV.