Three perfect gases at absolute temperatures \mathrm{T}_{1},\ \mathrm{T}_{2} and \mathrm{T}_{3} are mixed. The masses of molecules are \mathrm{m}_{1},\ \mathrm{m}_{2} and \mathrm{m}_{3} and the number of molecules are \mathrm{n}_{1},\ \mathrm{n}_{2} and \mathrm{n}_{3} respectively. Assuming no loss of energy, the final temperature of the mixture is:

Number of moles of first gas =\displaystyle \frac{\mathrm{n}_{1}}{\mathrm{N}_{\mathrm{A}}} Number of moles of second gas =\displaystyle \frac{\mathrm{n}_{2}}{\mathrm{N}_{\mathrm{A}}} Number of moles of third gas =\displaystyle \frac{\mathrm{n}_{3}}{\mathrm{N}_{\mathrm{A}}} If no loss of energy then \mathrm{P}_{1}\mathrm{V}_{1}+\mathrm{P}_{2}\mathrm{V}_{2}+\mathrm{P}_{3}\mathrm{V}_{3}= PV \displaystyle \frac{\mathrm{n}_{1}}{\mathrm{N}_{\mathrm{A}}}\mathrm{R}\mathrm{T}_{1}+\frac{\mathrm{n}_{2}}{\mathrm{N}_{\mathrm{A}}}\mathrm{R}\mathrm{T}_{2}+\frac{\mathrm{n}_{3}}{\mathrm{N}_{\mathrm{A}}}\mathrm{R}\mathrm{T}_{3} \displaystyle=\frac{\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}}{\mathrm{N}_{\mathrm{A}}}\mathrm{R}\mathrm{T}_{mix} \implies \displaystyle \mathrm{T}_{mix}=\frac{\mathrm{n}_{1}\mathrm{T}_{1}+\mathrm{n}_{2}\mathrm{T}_{2}+\mathrm{n}_{3}\mathrm{T}_{3}}{\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}} .

An ideal gas goes through a reversible cycle a\rightarrow b\rightarrow c\rightarrow d has the V- T diagram shown below. Process d\rightarrow a and b\rightarrow c are adiabatic.The corresponding P-V diagram for the process is ( all figure are schematic and not drawn to scale).

In process a \rightarrow b and c \rightarrow d the volume is directly proportional to temperature of gas, hence the process is isobaric, however, the slope of curve is different in each case. using ideal gas equation PV=nRT \Rightarrow V=\dfrac{nR}{P} T slope m=\dfrac{nR}{P} for a high pressure the slope of the V-T curve will be less. In given diagram the slope of c \rightarrow d process is lower than a \rightarrow b process, hence process a \rightarrow b is at higher pressure, from which option B and D can be ruled out.Now initial rate of change of volume w.r.t. temperature is very less from b \rightarrow c and d \rightarrow a as shown in V-T diagram.hence slope of P-V curve should tend to \lim_{\Delta V\rightarrow 0}\dfrac{\partial P }{\partial v}=\infty as initial change in volume is nearly zero, which can be verified in option AHence correct answer is option A.

Two moles of an ideal gas with \dfrac{C_P}{C_V}=\dfrac{5}{3} are mixed with 3 moles of another ideal gas with \dfrac{C_P}{C_V}=\dfrac{4}{3} . The value of \dfrac{C_P}{C_V} for the mixture is :

Let n_1,n_2 be the number of moles of gases 1 and 2 .We have \dfrac{C_{p_1}}{C_{v_1}}=\dfrac{5}{3} and We know C_{p_1}=C_{v_1}+R \therefore \dfrac{5C_{v_1}}{3}=C_{v_1}+R,\dfrac{2C_{v_1}}{3}R_1,C_{v_1}=\dfrac{3R}{2} A and C_{p_1}=C_{ v_ 1 }+R=\dfrac{5R}{2} Also, \dfrac{C_{ p_ 2}}{C_{ v_ 2 }}=\dfrac{4}{3} and C_{p_2}=C_{v_2}+R \therefore \dfrac{4C_{ v_ 2 }}{3}=C_{ v_ 2 }+R, \dfrac{C_{ v_ 2 }}{3}=R,C_{ v_ 2 }=3R C_{ p_ 2}=C_{ v_ 2 }+ R=4R Now C_{ v_ {mix} }=\dfrac{n_1C_{ v_ 1 }+n_2C_{ v_ 2 }}{n_1+n_2}=\dfrac{2\times \dfrac{3R}{2}+3\times 3R}{2+3} =\dfrac{3R+9R}{5}=\dfrac{12R}{5} C_{ p_ {mix} }=\dfrac{n_1C_{ p_ 1 }+n_2C_{ p_ 2 }}{n_1+n_2}=\dfrac{2\times \dfrac{5R}{2}+3\times 4R}{2+3} =\dfrac{17R}{5} l_{mix}=\dfrac{C_{ p_ {mix} }}{C_{ v_ {mix} }}=\dfrac{17}{12}=1.4166=1.42 Option (C) is correct.

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Thermal Properties Of Matter

Q: A leak proof cylinder of length 1m , made of a metal which has low coefficient of expansion is floating vertically in water at 0^oC such that its height above the water surface is 20cm . When the temperature of water is increased to 4^oC , the height of the cylinder above the water surface becomes 21cm . The density of water at T = 4^oC , relative to the density at T = 0^oC is close to:

Let A be the area of cross section of the cylinder. p_0 be the density of water at 0^oC p_4 be the density of water at 4^oC .Total length of the cylinder is 100cm Given that the length of the cylinder above the surface of water is 20cm At 0^oC , the weight of the cylinder is balanced by the buoyant force. p_0A \times 0.8 \times g = w [Buoyant force = pA lg ] ...(i)At 4^oC , thef weight of the cylinder is balanced by the buoyant force. p_4A\times 0.79\times g = w ...(ii)Dividing (i) by (ii) \dfrac{p_0\times 0.8}{p_4 \times 0.79} = 1 , p_4 = \dfrac{p_0\times 0.8}{0.79} = 0.012p_0 \dfrac{p_4}{p_0} = 1.012 \simeq 1.01

Q: Which of the following shows the correct relationship between the pressure 'P' and density \rho of an ideal gas at constant temperature ?

From ideal gas law: PV=nRT P=\cfrac{MnRT}{MV}, \quad M: Molecular mass number But \rho=Mn/V P=\rho RT/M At constant temperature, P \propto \rho Hence, the correct graph is (d).

Q: 100 \mathrm{g} of water is heated from 30^{\mathrm{o}}\mathrm{C} to 50^{\mathrm{o}}\mathrm{C} . Ignoring the slight expansion of the water, the change in its internal energy is (specific heat of water is 4184 \mathrm{J}/\mathrm{K}\mathrm{g}/\mathrm{K} )

\Delta Q= \Delta U +\Delta W Since \Delta W =0 , \Delta Q= \Delta U \Delta U=ms \Delta T= 0.1\times 4.184 \times 20=8.368 kJ

Physics
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JEE Mains JEE Advanced NEET BITSAT Easy Qs Med Qs Hard Qs

Three perfect gases at absolute temperatures $T_{1}, T_{2}$ and $T_{3}$ are mixed. The masses of molecules are $m_{1}, m_{2}$ and $m_{3}$ and the number of molecules are $n_{1}, n_{2}$ and $n_{3}$ respectively. Assuming no loss of energy, the final temperature of the mixture is:

\frac{( T _{1} + T _{2} + T _{3} )}{3}

\frac{n _{1} T _{l} + n _{2} T _{2} + n _{3} T _{3}}{n _{1} + n _{2} + n _{3}}

\frac{n _{1} T _{1}^{2} + n _{2} T _{2}^{2} + n _{3} T _{3}^{3}}{n _{1} T _{1} + n _{2} T _{2} + n _{3} T _{3}}

\frac{n _{1}^{2} T _{1}^{2} + n _{2}^{2} T _{2}^{2} + n _{3} T _{3}^{3}}{n _{1} T _{1} + n _{2} T _{2} + n _{3} T _{3}}

Medium

JEE Mains

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An ideal gas goes through a reversible cycle $a \to b \to c \to d$ has the V- T diagram shown below. Process $d \to a$ and $b \to c$ are adiabatic.
The corresponding P-V diagram for the process is ( all figure are schematic and not drawn to scale).

Hard

JEE Mains

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A leak proof cylinder of length $1 m$ , made of a metal which has low coefficient of expansion is floating vertically in water at $0^{o} C$ such that its height above the water surface is $20 c m$ . When the temperature of water is increased to $4^{o} C$ , the height of the cylinder above the water surface becomes $21 c m$ . The density of water at $T = 4^{o} C$ , relative to the density at $T = 0^{o} C$ is close to:

1.26

1.03

1.04

1.01

Medium

JEE Mains

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Which of the following shows the correct relationship between the pressure $^{'} P^{'}$ and density $ρ$ of an ideal gas at constant temperature ?

Medium

JEE Mains

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$100$ $g$ of water is heated from $3 0^{o} C$ to $5 0^{o} C$ . Ignoring the slight expansion of the water, the change in its internal energy is (specific heat of water is $4184$ $J / K g / K$ )

4.2

k J

8.4

k J

84

k J

2.1

k J

Medium

JEE Mains

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Two moles of an ideal gas with $\frac{C _{P}}{C _{V}} = \frac{5}{3}$ are mixed with $3$ moles of another ideal gas with $\frac{C _{P}}{C _{V}} = \frac{4}{3}$ . The value of $\frac{C _{P}}{C _{V}}$ for the mixture is :

1.47

1.45

1.42

1.50

Medium

JEE Mains

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Thermal Properties Of Matter

Physics 536 Views

Three perfect gases at absolute temperatures T1​, T2​ and T3​ are mixed. The masses of molecules are m1​, m2​ and m3​ and the number of molecules are n1​, n2​ and n3​ respectively. Assuming no loss of energy, the final temperature of the mixture is:

An ideal gas goes through a reversible cycle a→b→c→d has the V- T diagram shown below. Process d→a and b→c are adiabatic. The corresponding P-V diagram for the process is ( all figure are schematic and not drawn to scale).

Which of the following shows the correct relationship between the pressure ′P′ and density ρ of an ideal gas at constant temperature ?

100 g of water is heated from 30oC to 50oC. Ignoring the slight expansion of the water, the change in its internal energy is (specific heat of water is 4184 J/Kg/K)

Two moles of an ideal gas with CV​CP​​=35​ are mixed with 3 moles of another ideal gas with CV​CP​​=34​. The value of CV​CP​​ for the mixture is :

Physics
536 Views

Three perfect gases at absolute temperatures $T_{1}, T_{2}$ and $T_{3}$ are mixed. The masses of molecules are $m_{1}, m_{2}$ and $m_{3}$ and the number of molecules are $n_{1}, n_{2}$ and $n_{3}$ respectively. Assuming no loss of energy, the final temperature of the mixture is:

An ideal gas goes through a reversible cycle $a \to b \to c \to d$ has the V- T diagram shown below. Process $d \to a$ and $b \to c$ are adiabatic.
The corresponding P-V diagram for the process is ( all figure are schematic and not drawn to scale).

Which of the following shows the correct relationship between the pressure $^{'} P^{'}$ and density $ρ$ of an ideal gas at constant temperature ?

$100$ $g$ of water is heated from $3 0^{o} C$ to $5 0^{o} C$ . Ignoring the slight expansion of the water, the change in its internal energy is (specific heat of water is $4184$ $J / K g / K$ )

Two moles of an ideal gas with $\frac{C _{P}}{C _{V}} = \frac{5}{3}$ are mixed with $3$ moles of another ideal gas with $\frac{C _{P}}{C _{V}} = \frac{4}{3}$ . The value of $\frac{C _{P}}{C _{V}}$ for the mixture is :