In given figure, an adiabatic cylindrical tube of volume 2 V 0 is divided in two equal parts by a frictionless adiabatic separator. An ideal gas in left side of a tube having pressure P 1 and temperature T 1 , where as on the right side having pressure P 2 and temperature T 2 . C p C v = γ is the same for both the gases. The separator is slid slowly and is released at a position where it can stay in equilibrium. Find (a) the final volumes of the two parts, (b) the heat given to the gas in the left part and (c) the final common pressure of the gases.

(a).Let volume of the left and right part be V 1 & V 2 , pressure P at equilibrium. It is an adiabatic cylindrical tube therefore no exchange of heat from surroundings. For left part P 1 V 0 γ = P V 1 γ . . . ( 1 ) , for right part P 2 V 0 γ = P V 2 γ . . . ( 2 ) Dividing ( 1 ) by ( 2 ) , V 1 = P 1 P 2 1 γ V 2 V 1 + V 2 = 2 V 0 P 1 P 2 1 γ V 0 + V 2 = 2 V 0 ⇒ V 2 = 2 P 2 1 γ V 0 P 1 1 γ + P 2 1 γ , V 1 = 2 P 1 1 γ V 0 P 1 1 γ + P 2 1 γ (b). Adiabatic cylindrical tube has no exchange of heat from surroundings, therefore heat given to the left part is zero. (c). Putting the value of V 1 in equation ( 1 ) , P = P 1 1 γ + P 2 1 γ γ 2 γ

Cloud formation condition Consider a simplified model of cloud formation. Hot air in contact with the earth’s surface contains water vapour. This air rises connectively till the water vapour content reaches its saturation pressure. When this happens, the water vapour starts condensing and droplets are formed. We shall estimate the height at which this happens. We assume that the atmosphere consists of the diatomic gases oxygen and nitrogen in the mass proportion 21 : 79 respectively. We further assume that the atmosphere is an ideal gas, g the acceleration due to gravity is constant and air processes are adiabatic. Under these assumptions one can show that the pressure is given by p = p 0 T 0 - τ Z T 0 α Here p 0 and T 0 is the pressure and temperature respectively at sea level ( z = 0 ) , τ is the lapse rate (magnitude of the change in temperature T with height z above the earth’s surface, i.e. τ > 0 ). (a) Obtain an expression for the lapse rate Γ in terms of γ , R , g and m a . Here γ is the ratio of specific heat at constant pressure to specific heat at constant volume; R , the gas constant; and m a , the relevant molar mass. (b) Estimate the change in temperature when we ascend a height of one kilometre ? (c) Show that pressure will depend on height as given by Eq. (1). Find an explicit expression for exponent α in terms of γ . (d) According to this model what is the height to which the atmosphere extends? Take T 0 = 300 K and p 0 = 1 atm .

A According to question, Lapse rate τ = d T d z For an adiabatic process, P T γ 1 - γ = Constant Differentiate the above expression w.r.t z ⇒ P d T γ 1 - γ d z + T γ 1 - γ d P d z = 0 ⇒ P d T γ 1 - γ d T d T d z + T γ 1 - γ d P d z = 0 ⇒ P γ 1 - γ T γ 1 - γ - 1 d T d z + T γ 1 - γ d P d z = 0 ⇒ T γ 1 - γ P γ 1 - γ 1 T d T d z + d P d z = 0 ⇒ P γ 1 - γ 1 T d T d z + d P d z = 0 ⇒ P γ 1 - γ 1 T d T d z - ρ g = 0 d P d z = - ρ g ⇒ P γ 1 - γ 1 T d T d z = ρ g . . . . 2 Using Ideal gas equation, P V = n R T P m a = ρ R T ⇒ P T = ρ R m a . . . . 3 From 2 and 3 , we get ⇒ ρ R m a γ 1 - γ d T d z = ρ g ⇒ d T d z = m a g 1 - γ R γ = m a g γ - 1 R γ ⇒ τ = m a g γ - 1 R γ . . . . 4 B From 4 , ∆ T ∆ z = m a g γ - 1 Rγ Molar mass of the gas mixture is m a = 21 32 + 79 28 100 = 28 . 84 × 10 - 3 kg mol - 1 As the air is mixture of two diatomic gases, hence, the γ = 7 5 . ∆ T = 28 . 84 × 10 - 3 10 7 5 - 1 8 . 314 7 5 1000 = 9 . 91 K Hence, temperature change is 9 . 91 K . C From 1 , τ = ∆ T ∆ z = T - T 0 z - 0 = T 0 - T z ⇒ T = T 0 - τ z For adiabatic process, P T γ 1 - γ = Constant ⇒ P T γ 1 - γ = P 0 T 0 γ 1 - γ ⇒ P = P 0 T 0 T γ 1 - γ = P 0 T 0 T 0 - T z γ 1 - γ = P 0 T 0 - T z T 0 1 - γ γ Hence, α = 1 - γ γ . D We know that temperature of atmosphere is decreasing with height. Minimum temperature possible is 0 K . From T = T 0 - τ z ⇒ 0 = T 0 - τ z ⇒ z = T 0 τ From previous calculations, we know that τ is 9 . 91 K km - 1 . ⇒ z = 300 9 . 91 = 30 . 27 ≈ 30 . 3 km

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A weightless piston divides a thermally insulated cylinder into two parts of volumes $V$ and $3 V . 2 moles$ of an ideal gas at pressure $P = 2$ atmosphere are confined to the part with volume $V = 1 liter$ . The remainder of the cylinder is evacuated. The piston is now released and the gas expands to fill the entire space of the cylinder. The piston is then pressed back to the initial position. Find the increase of internal energy in the process and final temperature of the gas. The ratio of the specific heats of the gas, $γ = 1.5 .$

Important Questions on Thermodynamics

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Alpha Question Bank for Engineering: Physics>Chapter 17 - Thermodynamics>Exercise - 4>Q 14

Two containers

A

and

B

of equal volume

\frac{V_{0}}{2}

each are connected by a narrow tube which can be closed by a valve. The containers are fitted with pistons which can be moved to change the volumes. Initially, the valve is open and the containers contain an ideal gas

(\frac{C_{p}}{C_{v}} = γ)

at atmospheric pressure

P_{0}

and atmospheric temperature

2 T_{0}

. The walls of the containers

A

are highly conducting and of

B

are non-conducting. The valve is now closed and the pistons are slowly pulled out to increase the volumes of the containers to double the original value. (a) Calculate the temperatures and pressures in the two containers. (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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Alpha Question Bank for Engineering: Physics>Chapter 17 - Thermodynamics>Exercise - 4>Q 15

In given figure, an adiabatic cylindrical tube of volume $2 V_{0}$ is divided in two equal parts by a frictionless adiabatic separator. An ideal gas in left side of a tube having pressure $P_{1}$ and temperature $T_{1}$ , where as on the right side having pressure $P_{2}$ and temperature $T_{2} . \frac{C_{p}}{C_{v}} = γ$ is the same for both the gases. The separator is slid slowly and is released at a position where it can stay in equilibrium. Find (a) the final volumes of the two parts, (b) the heat given to the gas in the left part and (c) the final common pressure of the gases.

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Alpha Question Bank for Engineering: Physics>Chapter 17 - Thermodynamics>Exercise - 4>Q 16

A cylindrical tube with adiabatic walls having volume $2 V_{0}$ contains an ideal monoatomic gas as shown in figure. The tube is divided into two equal parts by a fixed super conducting wall. Initially, the pressure and the temperature are $P_{0}, T_{0}$ on the left and $2 P_{0}, 2 T_{0}$ on the right. When system is left for sufficient amount of time the temperature on both sides becomes equal (a) Find work done by the gas on the right part? (b) Find the final pressures on the two sides. (c) Find the final equilibrium temperature. (d) How much heat has flown from the gas on the right to the gas on the left?

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Alpha Question Bank for Engineering: Physics>Chapter 17 - Thermodynamics>Exercise - 4>Q 17

An ideal gas $(\frac{C_{p}}{C_{v}} = γ)$ having initial pressure $P_{0}$ and volume $V_{0}$ .

(a) The gas is taken isothermally to a pressure $2 P_{0}$ and then adiabatically to a pressure $4 P_{0}$ . Find the final volume.

(b) The gas is brought back to its initial state. It is adiabatically taken to a pressure $2 P_{0}$ and then isothermally to a pressure $4 P_{0}$ . Find the final volume.

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Alpha Question Bank for Engineering: Physics>Chapter 17 - Thermodynamics>Exercise - 4>Q 18

Two samples

A

and

B

of the same gas have equal volumes and pressures. The gas in sample

A

is expanded isothermally to four times of its initial volume and the gas in

B

is expanded adiabatically to double of its volume. If works done in isothermal process is twice that of adiabatic process, then show that

γ

satisfies the equation

1 - 2^{(1 - γ)} = (γ - 1) In 2 .

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Alpha Question Bank for Engineering: Physics>Chapter 17 - Thermodynamics>Exercise - 4>Q 19

A Carnot engine cycle is shown in the figure (2). The cycle runs between temperatures $T_{H} = α T_{0} and T_{L} = T_{0} (α > 1) .$ Minimum and maximum volume at state $1$ and state $3$ are $V_{0}$ and $n V_{0}$ , respectively. The cycle uses one mole of an ideal gas with $\frac{C_{P}}{C_{V}} = γ$ . Here $C_{P}$ and $C_{V}$ are the specific heats at constant pressure and volume respectively. You must express all answers in terms of the given parameters $\{α, n, T_{0}, V_{0}\}, ?$ and universal gas constant $R$ .

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(a) Find $P, V, T$ for all the states

(b) Calculate the work done by the engine in each process $W_{12}, W_{23}, W_{34}, W_{41} .$

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Alpha Question Bank for Engineering: Physics>Chapter 17 - Thermodynamics>Exercise - 4>Q 20

Cloud formation condition

Consider a simplified model of cloud formation. Hot air in contact with the earth’s surface contains water vapour. This air rises connectively till the water vapour content reaches its saturation pressure. When this happens, the water vapour starts condensing and droplets are formed. We shall estimate the height at which this happens. We assume that the atmosphere consists of the diatomic gases oxygen and nitrogen in the mass proportion $21 : 79$ respectively. We further assume that the atmosphere is an ideal gas, $g$ the acceleration due to gravity is constant and air processes are adiabatic. Under these assumptions one can show that the pressure is given by

$p = p_{0} {(\frac{T_{0} - τ Z}{T_{0}})}^{α}$

Here $p_{0}$ and $T_{0}$ is the pressure and temperature respectively at sea level $(z = 0), τ$ is the lapse rate (magnitude of the change in temperature $T$ with height $z$ above the earth’s surface, i.e. $τ > 0$ ).

(a) Obtain an expression for the lapse rate $Γ$ in terms of $γ, R, g$ and $m_{a}$ . Here $γ$ is the ratio of specific heat at constant pressure to specific heat at constant volume; $R$ , the gas constant; and $m_{a}$ , the relevant molar mass.

(b) Estimate the change in temperature when we ascend a height of one kilometre ?

(c) Show that pressure will depend on height as given by Eq. (1). Find an explicit expression for exponent $α$ in terms of $γ$ .

(d) According to this model what is the height to which the atmosphere extends? Take $T_{0} = 300 K$ and $p_{0} = 1 atm .$

Important Questions on Thermodynamics

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