In a cylinder, 2 . 0 moles of an ideal monatomic gas initially at 1 . 0 × 10 6 Pa and 300 K expands until its volume doubles. Compute the work done if the expansion is adiabatic?

Let P 1 = 1 × 10 6 Pa = 10 6 N / m 2 ; T 1 = 300 K ; n = 2 moles. Final volume = 2 (initial volume) ⇒ V 2 = 2 V 1 W adiabatic = n R T 1 - T 2 γ - 1 For adiabatic process: T 1 V 1 γ - 1 = T 2 V 2 γ - 1 T 2 = T 1 V 1 V 2 γ - 1 = 300 × 1 2 5 / 3 - 1 ⇒ T 2 = 189 K W = n R T 1 - T 2 γ - 1 = 2 × 8 . 3 × 300 - 189 5 / 3 - 1 ⇒ W = 2764 J Thus work done is 2764 J .

In a cylinder, 2 . 0 moles of an ideal monatomic gas initially at 1 . 0 × 10 6 Pa and 300 K expands until its volume doubles. Compute the work done if the expansion is adiabatic?

Let P 1 = 1 × 10 6 Pa = 10 6 N / m 2 ; T 1 = 300 K ; n = 2 moles. Final volume = 2 (initial volume) ⇒ V 2 = 2 V 1 W adiabatic = n R T 1 - T 2 γ - 1 For adiabatic process: T 1 V 1 γ - 1 = T 2 V 2 γ - 1 T 2 = T 1 V 1 V 2 γ - 1 = 300 × 1 2 5 / 3 - 1 ⇒ T 2 = 189 K W = n R T 1 - T 2 γ - 1 = 2 × 8 . 3 × 300 - 189 5 / 3 - 1 ⇒ W = 2764 J Thus work done is 2764 J .

In a cylinder, 2 . 0 moles of an ideal monatomic gas initially at 1 . 0 × 10 6 Pa and 300 K expands until its volume doubles. Compute the work done if the expansion is isobaric.

It is said in the question that 2 . 0 moles of ideal monoatomic gas expands untill its volume doubles. Pressure given is 1 . 0 × 10 6 Pa . We are supposed to find out work done if expansion is isobaric. Let P 1 = 1 × 10 6 Pa = 10 6 N / m 2 ; T 1 = 300 K ; n = 2 moles. Final volume = 2 (initial volume) ⇒ V 2 = 2 V 1 W isobaric = P V 2 - V 1 ⇒ P 1 V 1 = n R T 1 V 1 = n R T 1 P 1 = 2 × 8 . 3 × 300 10 6 ⇒ V 1 = 4 . 98 × 10 - 3 m 3 W = P V 2 - V 1 = P 2 V 1 - V 1 = P V 1 ⇒ W = 10 6 × 4 . 98 × 10 - 3 J ⇒ W = 4980 J Thus work done is 4890 J

The heat supplied to one mole of an ideal monatomic gas in increasing temperature from T 0 to 2 T 0 is 2 R T 0 . Find the equation of process to which the gas follows.

The heat supplied to the gas, Q = n C Δ T Here, n is the number of moles and C is the molar heat capacity of the gas. In this case, n = 1 . For polytropic process molar heat capacity is given by, C = R γ - 1 + R 1 - N = R 5 / 3 - 1 + R 1 - N = 3 2 R + R 1 - N Q = n 3 R 2 + R ( 1 - N ) 2 T 0 - T 0 = 1 R T 0 5 - 3 N 2 1 - N = 2 R T 0 (given) or 5 - 3 N = 4 1 - N or N = - 1 Hence, the equation of process will be: P V - 1 = constant

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In a cylinder, $2.0$ moles of an ideal monatomic gas initially at $1.0 \times 10^{6} Pa$ and $300 K$ expands until its volume doubles. Compute the work done if the expansion is adiabatic?

Important Questions on Thermodynamics

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PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 4 - Thermodynamics>ILLUSTRATIONS>Q 4.23

In a cylinder,

2.0

moles of an ideal monatomic gas initially at

1.0 \times 10^{6} Pa

and

300 K

expands until its volume doubles. Compute the work done if the expansion is isobaric.

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PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 4 - Thermodynamics>ILLUSTRATIONS>Q 4.24

One mole of a perfect gas, initially at a pressure and temperature of $10^{5} N / m^{2}$ and $300 K$ , respectively, expands isothermally until its volume is doubled and then adiabatically until its volume is again doubled. Find the final pressure and temperature of the gas. Find the total work done during the isothermal and adiabatic processes. Given $γ = 1.4$ . Also draw the $P - V$ diagram for the process.

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PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 4 - Thermodynamics>ILLUSTRATIONS>Q 4.25

A cylindrical container having nonconducting walls is partitioned in two equal parts such that the volume of the each parts is equal to $V_{0}$ . A movable nonconducting piston is kept between the two parts. Gas on left is slowly heated so that the gas on right is compressed upto volume $V_{0} / 8$ . Find pressure and temperature on both sides if initial pressure and temperature, were $P_{0}$ and $T_{0}$ respectively. Also find heat given by the heater to the gas. (number of moles in each part is $n$ )

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PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 4 - Thermodynamics>ILLUSTRATIONS>Q 4.26

An adiabatic vertical cylinder fitted with and adiabatic frictionless piston contains $n$ moles of an ideal diatomic gas. The initial volume and temperature of the gas are $V_{0}$ and $T_{0}$ , respectively, with the piston fixed tightly. The atmospheric pressure is $P_{0}$ . If the piston be released, find the equilibrium temperature and volume of the gas, if the heat capacity of the cylinder and piston is negligibly small. Weight of the piston and cross-sectional area of the cylinder are $W$ and $A$ . respectively.

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PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 4 - Thermodynamics>ILLUSTRATIONS>Q 4.27

Three moles of an ideal monoatomic gas are confined within a cylinder by a massless spring loaded with a frictionless piston of negligible mass and of cross - sectional area

4 \times 10^{- 3} m^{2}

. The spring is initially in its relaxed state. Now the gas is heated by a heater for some time. During this time the gas expands and does

50 J

of work in moving the piston through a distance of

0.1 m

. The temperature increased by

50 K

. Calculate the spring constant and the heat supplied by the heater. (Given: Atmospheric pressure

P_{0} = 10^{5} N / m^{2}

and gas constant

R = \frac{25}{3} J / mol - K

)

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PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 4 - Thermodynamics>ILLUSTRATIONS>Q 4.28

An ideal gas

(C_{p} / C_{V} = γ)

is taken through a process in which the pressure and the volume vary as

p = a V^{b}

. Find the value of

b

for which the specific heat capacity in the process is zero.

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PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 4 - Thermodynamics>ILLUSTRATIONS>Q 4.29

The heat supplied to one mole of an ideal monatomic gas in increasing temperature from

T_{0}

2 T_{0}

2 R T_{0}

. Find the equation of process to which the gas follows.

HARD

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PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 4 - Thermodynamics>ILLUSTRATIONS>Q 4.30

One mole of an ideal gas with adiabatic exponent

γ

whose pressure changes with volume as

P = α V

, where

α

is a constant, is expanded so that its volume increase

η

times. Find the change in internal energy and heat capacity of the gas. Initial volume of gas is

V_{0}

In a cylinder, 2.0 moles of an ideal monatomic gas initially at 1.0×106 Pa and 300 K expands until its volume doubles. Compute the work done if the expansion is adiabatic?

Important Questions on Thermodynamics

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In a cylinder, $2.0$ moles of an ideal monatomic gas initially at $1.0 \times 10^{6} Pa$ and $300 K$ expands until its volume doubles. Compute the work done if the expansion is adiabatic?