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The ratio of the molar heat capacities of an ideal gas is CP/CV = 7/6. Calculate the change in internal energy of 1.0 mole of the gas when its temperature is raised by 50 K
(a) keeping the pressure constant,
(b) keeping the volume constant and
n = number of moles = 1,
Cv = specific heat capacity at constant volume,
Cp = specific heat capacity at constant pressure
dT = change in temperature = 50K.
γ= Ratio of molar heat capacities = CP/CV = 7/6 => Cv = 6Cp/7.
(a) Formula used:
Pressure constant: Isobaric process. For an isobaric process,
change in internal energy dU = nCvdT,
n = number of moles,
Cv = specific heat at constant volume,
dT = rise in temperature
Also, Cp-Cv = R.
Cp = specific heat at constant pressure
Cv = specific heat at constant volume
R = universal gas constant = 8.314 J/mol/K
Substituting: Cp - 6Cp/7 = Cp/7 = R => Cp = 7R.
Therefore Cv = Cp - R = 6R = (6 X 8.314)J/mol/K
dU = 1 mol X (6 X 8.314)J/mol/K X 50K = 2494.2 J(Ans)
(b) Volume constant: Isochoric process, dV = 0(change in volume)
First law of thermodynamics gives us: dU = dQ - dW
Where dU = change in internal energy, dQ = change in heat,
dW = work done = Pressure x change in volume = PdV
Since dV = 0, dU = dQ.
Hence, dU = nCvdT since dQ = nCvdT.
n = Number of moles,
Cv = Specific heat at constant volume,
dT =Change in temperature
Putting the values in the above formula, we get
Therefore, dU = 1 mol x (6 x 8.314) J/mol/K x 50K
= 2494.2 J(Ans)
(c) Adiabatic process: dQ(heat change) = 0. Therefore,
dU(change in internal energy) = dW(work done)
Since dQ = dU + dW.
For an adiabatic process, dW = dT/(
Figure shows a cylindrical container containing oxygen (γ = 1.4) and closed by a 50 kg frictionless piston. The area of cross section is 100 cm2, atmospheric pressure is 100 kPa and g is 10 ms–2. The cylinder is slowly heated for some time. Find the amount of heat supplied to the gas if the piston moves out through a distance of 20 cm.
A sample of air weighing 1.18g occupies 1.0 × 103 cm3 when kept at 300K and 1.0 × 105 Pa. When 2.0 cal of heat is added to it at constant volume, its temperature increases by 1°C. Calculate the amount of heat needed to increase the temperature of air by 1°C at constant pressure if the mechanical equivalent of heat is 4.2 × 107 erg cal–1. Assume that air behaves as an ideal gas.
An ideal gas expands from 100 cm3 to 200 cm3 at a constant pressure of 2.0 × 105 Pa when 50J of heat is supplied to it. Calculate
(a) the change in internal energy of the gas.
(b) the number of moles in the gas if the initial temperature is 300K.
(c) the molar heat capacity CP at constant pressure and
(d) the molar heat capacity CV at constant volume.
An ideal gas (γ = 1.67) is taken through the process abc shown in figure. The temperature at the point a is 300K. Calculate
(a) the temperature at b and c,
(b) the work done in the process,
(c) the amount of heat supplied in the path ab and in the path bc and
(d) the change in the internal energy of the gas in the process.
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A gas is enclosed in a cylindrical can fitted with a piston. The walls of the can and the piston are adiabatic. The initial pressure, volume and temperature of the gas are 100 kPa, 400 cm3 and 300 K respectively. The ratio of the specific heat capacities of the gas is CP/CV = 1.5. Find the pressure and the temperature of the gas if it is
(a) suddenly compressed
(b) slowly compressed to 100 cm3.
The initial pressure and volume of a given mass of a gas (CP/CV = γ) are P0 and V0. The gas can exchange heat with the surrounding.
(a) It is slowly compressed to a volume V0/2 and then suddenly compressed to V0/4. Find the final pressure.
(b) If the gas is suddenly compressed form the volume V0 to V0/2 and then slowly compressed to V0/4, what will be the final pressure?
Consider a given sample of an ideal gas (CP/CV = γ) having initial pressure P0 and volume V0.
(a) The gas is isothermally taken to a pressure P0/2 and from there adiabatically to a pressure P0/4. Find the final volume.
(b) The gas is brought back to its initial state. It is adiabatically taken to a pressure P0/2 and from there isothermally to a pressure P0/4. Find the final volume.
A sample of an ideal gas (γ = 1.5) is compressed adiabatically from a volume of 150 cm3 to 50 cm3. The initial pressure and the initial temperature are 150 kPa and 300 K. Find
(a) the number of moles of the gas in the sample,
(b) the molar heat capacity at constant volume,
(c) the final pressure and temperature,
(d) the work done by the gas in the process and
(e) the change in internal energy of the gas.
Two samples A and B of the same gas have equal volumes and pressures. The gas in sample A is expanded isothermally to double its volume and the gas in B is expanded adiabatically to double its volume. If the work done by the gas is the same for the two cases, show that γ satisfies the equation 1 – 21– γ = (γ – 1) ln2.
1 litre of an ideal gas (γ = 1.5) at 300 K is suddenly compressed to half its original volume.
(a) Find the ratio of the final pressure to the initial pressure.
(b) If the original pressure is 100 kPa, find the work done by the gas in the process.
(c) What is the change in internal energy?
(d) What is the final temperature?
(e) The gas is now cooled to 300 K keeping its pressure constant.
Calculate the work done during the process.
(f) The gas is now expanded isothermally to achieve its original volume of 1 litre. Calculate the work done by the gas.
(g) Calculate the total work done in the cycle.
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(a) Find the number of moles of the gas in each vessel.
(b) 5.0 J of heat is supplied to the gas in the vessel A and 10 J to the gas in the vessel B. Assuming no appreciable transfer of heat from A to B calculate the difference in the heights of mercury in the two sides of the manometer. Gas constant R = 8.3 J K–1 mol–1.
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(a) Find the temperatures and pressures in the two vessels.
(b) The valve is now opened for sufficient time so that the gases acquire a common temperature and pressure. Find the new values of the temperature and the pressure.
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(a) the volumes of the two parts,
(b) the heat given to the gas in the left part
(c) the final common pressure of the gases.
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