Three moles of-an ideal monatomic gas perform a cycle shown in Fig. The gas temperatures in different states are T 1 = 400 K , T 2 = 800 K , T 3 = 2400 K , and T 4 = 1200 K . Determine the work A done by the gas luring the cycle.

The figure shows that on segments, 1 - 2 and 3 - 4 , pressure is directly proportional to temperature. It follows from the equation of state for an ideal gas that the gas volume remains unchanged in this case, and the gas does no work. Therefore, we must find the work done only in isobaric processes, 2 - 3 and 4 - 1 . The work A 23 = p 2 V 3 - V 2 is done on segment 2 - 3 and A 41 = p 1 V 1 - V 4 on segment 4 - 1 . The total work A done by the gas during a cycle is, A = p 2 V 3 - V 2 + p 1 V 1 - V 4 The equation of state for three moles of the ideal gas can be written as p V = 3 R T , and hence p 1 V 1 = 3 R T 1 , p 1 V 4 = 3 R T 4 , p 2 V 2 = 3 R T 2 , p 2 V 3 = p 3 V 3 = 3 R T 3 . Substituting these values into the expression for work, we finally obtain A = 3 R T 1 + T 3 - T 2 - T 4 ⇒ A = 2 × 10 4 J = 20 kJ

Find the work done by an ideal gas during a closed cycle 1 → 4 → 3 → 2 → 1 shown in the figure if P 1 = 10 5 Pa , P 0 = 3 × 10 5 Pa , P 2 = 4 × 10 5 Pa , V 2 – V 1 = 10 litre and segments 4 - 3 and 2 - 1 of the cycle are parallel to the V -axis?

According to the above figure given in the question, the cycle 1 → 4 → 3 → 2 → 1 is in fact equivalent to two simple cycles 1 → 0 → 2 → 1 and 0 → 4 → 3 → 0 , the work done by the gas is determined by the area of the corresponding cycle on the P - V diagram. In the first cycle, the work is positive, while in the second cycle it is negative (the work is done on the gas). The work done in the first cycle can easily be calculated: W 1 = 2 P 0 - P 1 V 2 - V 1 , As regards the cycle 0 → 4 → 3 → 0 , the triangle on the P - V diagram corresponding to it is similar to the triangle corresponding to the first cycle, therefore, the work W 2 done in the second cycle will be, W 2 = - W 1 P 2 - P 0 2 P 0 - P 1 2 (The areas of similar triangles are to each other as the squares of the lengths of the corresponding elements, in our case, altitudes) The total work done W during the cycle 1 → 4 → 3 → 2 → 1 will therefore be, W = W 1 1 - P 2 - P 0 2 P 0 - P 1 2 ⇒ W = 2 P 0 - P 1 V 2 - V 1 1 - P 2 - P 0 2 P 0 - P 1 2 ⇒ W = 2 3 × 10 5 - 10 5 10 × 10 - 3 1 - 4 × 10 5 - 3 × 10 5 2 3 × 10 5 - 10 5 2 = 750 J

A gas takes part in two thermal processes in which it is heated from the same initial state to the same final temperature. The processes are shown on the p - V diagram by straight lines 1 - 3 and 1 - 2 (Fig.). Indicate the process in which the amount of heat supplied to the gas is larger.

According to the first law of thermodynamics, the amount of heat Δ Q 1 received by a gas going over from state 1 p 0 , V 0 to state 2 p 1 , V 1 (Fig.) is Δ Q 1 = Δ U 1 + W 1 , where Δ U 1 is the change in its internal energy, and A 1 is the work done by the gas, W 1 = p 0 + p 1 V 1 - V 0 2 , The area of the graph bounded by volume axis. As the gas goes over from state 1 to state 3 p 2 , V 2 (points 2 and 3 lie on the same isotherm), the following relations are fulfilled: Δ Q 2 = Δ U 2 + A 2 , W 2 = p 0 + p 2 V 2 - V 0 2 Since the final temperature of the gas in states 2 and 3 is the same, Δ U 1 = Δ U 2 . In order to find out in which process the gas receives a larger amount of heat, we must compare the works W 1 and W 2 : W 1 - W 2 = p 0 + p 1 V 1 - V 0 2 - p 0 + p 2 V 2 - V 0 2 = p 0 V 1 - p 0 V 2 + p 2 V 0 - p 1 V 0 2 < 0 since p 0 V 1 < p 0 V 2 and p 2 V 0 < p 1 V 0 . Consequently, W 2 > W 1 and Δ Q 2 > Δ Q 1 , i.e. the amount of heat received by the gas in the process 1 → 3 is larger.

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A cyclic process (cycle) $1 - 2 - 3 - 4 - 1$ consisting of two isobars $2 - 3$ and $4 - 1$ , isochor $1 - 2$ , and a certain process $3 - 4$ represented by a straight line on the $p - V$ diagram (Fig.) involves $n$ moles of an ideal gas. The gas temperatures in states $1, 2$ and $3$ are $T_{1}, T_{2}$ , and $T_{3}$ respectively, and points $2$ and $4$ lie on the same isotherm. Determine the work $A$ done by the gas during the cycle.

Important Questions on Heat and Molecular Physics

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Science for Everyone Aptitude test problems in Physics>Chapter 2 - Heat and Molecular Physics>Exercise>Q 2.4

Three moles of-an ideal monatomic gas perform a cycle shown in Fig. The gas temperatures in different states are

Question Image

$T_{1} = 400 K, T_{2} = 800 K, T_{3} = 2400 K$ , and $T_{4} = 1200 K$ .

Determine the work $A$ done by the gas luring the cycle.

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JEE Advanced

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Science for Everyone Aptitude test problems in Physics>Chapter 2 - Heat and Molecular Physics>Exercise>Q 2.5

Find the work done by an ideal gas during a closed cycle $1 \to 4 \to 3 \to 2 \to 1$ shown in the figure if $P_{1} = 10^{5} Pa, P_{0} = 3 \times 10^{5} Pa, P_{2} = 4 \times 10^{5} Pa, V_{2} - V_{1} = 10 litre$ and segments $4 - 3$ and $2 - 1$ of the cycle are parallel to the $V$ -axis?

Question Image

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Science for Everyone Aptitude test problems in Physics>Chapter 2 - Heat and Molecular Physics>Exercise>Q 2.6

A gas takes part in two thermal processes in which it is heated from the same initial state to the same final temperature.

Question Image

The processes are shown on the $p - V$ diagram by straight lines $1 - 3$ and $1 - 2$ (Fig.). Indicate the process in which the amount of heat supplied to the gas is larger.

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JEE Advanced

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Science for Everyone Aptitude test problems in Physics>Chapter 2 - Heat and Molecular Physics>Exercise>Q 2.7

A vessel of volume $V = 30 ml$ is separated into three equal parts by stationary semipermeable thin particles (Fig.). The

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left, middle, and right parts are filled with $m_{H_{2}} = 30 g$ of hydrogen, $m_{O_{2}} = 160 g$ of oxygen, and $m_{N_{2}} = 70 g$ of nitrogen respectively. The left partition lets through only hydrogen, while the right partition lets through hydrogen and nitrogen.

What will be the pressure in each part of the vessel after the equilibrium has been set in if the vessel is kept at a constant temperature $T = 300 K$ ?

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Science for Everyone Aptitude test problems in Physics>Chapter 2 - Heat and Molecular Physics>Exercise>Q 2.8

The descent module of a spacecraft approaches the surface of a planet along the vertical at a constant velocity, transmitting the data on outer pressure to the spacecraft. The time dependence of pressure (in arbitrary units) is shown in Fig. The data transmitted by the module after landing are: the temperature

T = 700 K

and the free-fall acceleration

g = 10 m s^{- 2}

. Determine (a) the velocity

v

of landing of the module if the atmosphere of the planet is known to consist of carbon dioxide

C O_{2}

, and (b) the temperature

T_{h}

at an altitude

h = 15 km

above the surface of the planet.

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JEE Advanced

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Science for Everyone Aptitude test problems in Physics>Chapter 2 - Heat and Molecular Physics>Exercise>Q 2.9

A vertical thermally insulated cylinder of volume $V$ contains $n$ moles of an ideal monatomic gas under a weightless piston.

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A load of mass $M$ is placed on the piston, as a result of which the piston is displaced by a distance $h$ .

Determine the final temperature $T_{1}$ of the gas established after the piston has been displaced if the area of the piston is $S$ and the atmospheric pressure is $p_{0}$ .

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JEE Advanced

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Science for Everyone Aptitude test problems in Physics>Chapter 2 - Heat and Molecular Physics>Exercise>Q 2.1

A vertical cylinder of cross-sectional area $S$ contains one mole of an ideal monatomic gas under a piston of mass $M$ . At a certain instant, a heater which transmits to a gas an amount of heat $q$ per unit time is switched on under the piston. Determine the established velocity $v$ of the piston under the condition that the gas pressure under the piston is constant and equal to $p_{0}$ , and the gas under the piston is thermally insulated.

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Science for Everyone Aptitude test problems in Physics>Chapter 2 - Heat and Molecular Physics>Exercise>Q 2.11

The product of pressure and volume

(p V)

of a gas does not change with volume at a constant temperature only provided that the gas is ideal. Will the product

p V

be higher or lower under a very strong compression of a gas if no assumption is made concerning the ideal nature of the gas?

Important Questions on Heat and Molecular Physics

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