Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given:
γ = 1.5
T = 300 K
Initial volume of the gas, V1 = 1 L
Final volume, V2 = `1/2` L
(a) The process is adiabatic because volume is suddenly changed; so, no heat exchange is allowed.
P1V1γ = P2V2γ
Or `"P"_2 ="P"_1("V"_1/"V"_2)^gamma = "P"_1(2)^gamma`
`"P"_2/"P"_1 = 2^1.5 = 2 sqrt2`
(b) P1 = 100 kPa = 105 Pa
and P2 = `2sqrt2` × 105 Pa
Work done by an adiabatic process,
`"W" = ("P"_1"V"_1 - "P"_2"V"_2)/(gamma - 1)`
`"W" = (10^5 xx 10^-3 -2sqrt2 xx 10^5 xx 1/2 xx 10^-3)/(1.5 -1)`
W = -82 J
(c) Internal energy,
dQ = 0, as it is an adiabatic process.
⇒ dU = − dW = − (− 82 J) = 82 J
(d)
Also, for an adiabatic process,
T1V1γ−1 = T2V2γ−1
`"T"_2 ="T"_1 ("V"_1/"V"_2)^(gamma -1)`
= 300 × (2)0.5
`= 300 xx sqrt 2 xx = 300 xx 1..4142`
T2= 424 K
(e) The pressure is kept constant.
The process is isobaric; so, work done = PΔV=nRdT.
Here, n = `("P""V")/("R""T") = (10^5 xx 10 ^-3)/("R" xx 300) = 1/(3"R")`
So, work done =`1/(3"R") xx "R" xx (300-424) = -41.4"J"`
As pressure is constant,
`"V"_1/"T"_1 = "V"_2/"T"_2 ... (1)`
`"V" _1 = "V"_2("T"_1)/"T"_2`
(f)Work done in an isothermal process,
`"W" = "n""R""T" "l""n" "V"_2/"V"_1`
= `1/(3"R") xx "R" xx "T" xx ln (2)`
= 100 × ln 2 = 100 × 1.039
= 103 J
(g) Net work done (using first law of thermodynamics)
= − 82 − 41.4 + 103
= − 20.4 J
APPEARS IN
संबंधित प्रश्न
The energy of a given sample of an ideal gas depends only on its
Which of the following quantities is zero on an average for the molecules of an ideal gas in equilibrium?
The average momentum of a molecule in a sample of an ideal gas depends on
Find the number of molecules in 1 cm3 of an ideal gas at 0°C and at a pressure of 10−5mm of mercury.
Use R = 8.31 J K-1 mol-1
Let Q and W denote the amount of heat given to an ideal gas and the work done by it in an isothermal process.
A rigid container of negligible heat capacity contains one mole of an ideal gas. The temperature of the gas increases by 1° C if 3.0 cal of heat is added to it. The gas may be
(a) helium
(b) argon
(c) oxygen
(d) carbon dioxide
The 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 m s−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.
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 (c) adiaba
An ideal gas at pressure 2.5 × 105 Pa and temperature 300 K occupies 100 cc. It is adiabatically compressed to half its original volume. Calculate (a) the final pressure (b) the final temperature and (c) the work done by the gas in the process. Take γ = 1.5
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.
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.
An ideal gas of density 1.7 × 10−3 g cm−3 at a pressure of 1.5 × 105 Pa is filled in a Kundt's tube. When the gas is resonated at a frequency of 3.0 kHz, nodes are formed at a separation of 6.0 cm. Calculate the molar heat capacities Cp and Cv of the gas.
A cubic vessel (with faces horizontal + vertical) contains an ideal gas at NTP. The vessel is being carried by a rocket which is moving at a speed of 500 ms–1 in vertical direction. The pressure of the gas inside the vessel as observed by us on the ground ______.
ABCDEFGH is a hollow cube made of an insulator (Figure). Face ABCD has positive charge on it. Inside the cube, we have ionized hydrogen. The usual kinetic theory expression for pressure ______.
- will be valid.
- will not be valid since the ions would experience forces other than due to collisions with the walls.
- will not be valid since collisions with walls would not be elastic.
- will not be valid because isotropy is lost.
In a diatomic molecule, the rotational energy at a given temperature ______.
- obeys Maxwell’s distribution.
- have the same value for all molecules.
- equals the translational kinetic energy for each molecule.
- is (2/3)rd the translational kinetic energy for each molecule.
We have 0.5 g of hydrogen gas in a cubic chamber of size 3 cm kept at NTP. The gas in the chamber is compressed keeping the temperature constant till a final pressure of 100 atm. Is one justified in assuming the ideal gas law, in the final state?
(Hydrogen molecules can be consider as spheres of radius 1 Å).
