Kinetic Theory Test

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Kinetic Theory Test - 78

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Question 1
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A closed cylindrical vessel contains $$N$$ moles of an ideal diatomic gas at a temperature $$T$$. On supplying heat, the temperature remains same, but $$n$$ moles get dissociated into atoms. The heat supplied is:

A
$$\dfrac{5}{2}(N - n)RT$$

B
$$\dfrac{5}{2}n RT$$

C
$$\dfrac{1}{2}n RT$$

D
$$\dfrac{3}{2}n RT$$

Solution

Solution:- (C) $$\cfrac{1}{2} nRT$$
Since the gas is enclosed in a vessel, during heating, volume of the gas remains constant.
$$\therefore W = 0$$
$$\therefore q = \Delta{U} ..... \left( 1 \right)$$

Now,
As we know that,
$$\Delta{U} = n {C}_{v} \Delta{T}$$

Initial internal energy of the gas is
$${U}_{1} = N \left( \cfrac{5}{2} R \right) T = \cfrac{5}{2} NRT$$

Since $$n$$ moles dissociate into atoms, therefore after heating, the vessel contains $$\left( N - n \right)$$ moles of diatomic gas and $$2n$$ moles of mono atomic gas. Hence the internal energy of the gas after heating will be equal to

$${U}_{2} = \left( N - n \right) \left( \cfrac{5}{2} R \right) T + 2n \left( \cfrac{3}{2} R \right) T$$

$$\Rightarrow { U}_{2} = \cfrac{1}{2} nRT + \cfrac{5}{2} NRT$$

$$\therefore \Delta{U} = {U}_{2} - {U}_{1}$$

$$\Rightarrow \Delta{U} = \left( \cfrac{1}{2} nRT + \cfrac{5}{2} NRT \right) - \cfrac{5}{2} NRT$$

$$\Rightarrow \Delta{U} = \cfrac{1}{2} nRT$$

Now from $${eq}^{n} \left( 1 \right)$$, we get

$$q = \cfrac{1}{2} nRT$$
Question 2
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The number of molecules in 1 cc of water is closed to

A
6 X $$10^{23}$$

B
22.4 X $$10^{24}$$

C
$$\frac{10^{23}}{3}$$

D
$$10^{23}$$

Solution
Question 3
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The pressure of an ideal gas veries according to the law $$P=P_{0}-AV^{2}$$ where $$P_{0}$$ and A are positive constants. What is the highest temperature that can be attained by the gas?

A
$$\dfrac{P_{0}}{nR}\sqrt{\dfrac{P_{0}}{A}}$$

B
$$\dfrac{P_{0}}{nR}\sqrt{\dfrac{P_{0}}{2A}}$$

C
$$\dfrac{2P_{0}}{nR}\sqrt{\dfrac{P_{0}}{2A}}$$

D
$$\dfrac{2P_{0}}{3nR}\sqrt{\dfrac{P_{0}}{3A}}$$

Solution
Question 4
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The heat capicity of liquid water at constant pressure, $$C_p$$ is $$18 $$ cals $$ deg^{-1} mol^{-1}$$. The value of heat capacity of water at constant volume, Cv is approximately:

A
$$18 cal deg^{-1} mol^{-1}$$

B
$$16 cal deg^{-1} mol^{-1}$$

C
$$10.8 cal deg^{-1} mol^{-1}$$

D
Cannot be predicted

Solution
Question 5
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$$C_{\upsilon }$$ values for monoatomic and diatomic gases respectively are:

A
$$\cfrac{R}{2},\cfrac{3R}{2}$$

B
$$\cfrac{3R}{2},\cfrac{5R}{2}$$

C
$$\cfrac{5R}{2},\cfrac{7R}{2}$$

D
$$\cfrac{3R}{2},\cfrac{3R}{2}$$
Question 6
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A diatomic gas is undergoing a process for which P versus V relation is given as $${ PV }^{ -\frac { 5 }{ 3 } }$$ = constant. The molar heat capacity of the gas for this process is:

A
$$\cfrac{5R}{3}$$

B
$$\cfrac{23R}{8}$$

C
$$\cfrac{17R}{4}$$

D
$$\cfrac{13R}{5}$$

Solution
Question 7
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The molar heat capacity of eater at constant pressure, C, is $$75JK^{-1}mol^{-1}$$. When 1.0KJ of heat is supplied to 100g of water which is free to expand, the increase in temperature of water is :

A
0.24K

B
2.4K

C
1.3K

D
0.13K

Solution
Question 8
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The heat capacity of liquid water is 75.6 J/mol k, while the enthaply of fusion of ice is 6.0 kJ/mol. What is the smallest number of ice cubes at $$0^{0}C$$ , each containing 9.0 g of water neede to cool 500 g of liquid water from $$20^{0}C$$ to $$0^{0}C$$?

A
1

B
7

C
14

D
None of these

Solution
Question 9
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Two closed containers of equal volume filled with air at pressure $$P_{0}$$ and temperature $$T_{0}$$. Both are connected by narrow tube. If one of the container is maintained at temperature $$T_{0}$$ and another at temperature T, then new pressure in the containers will be

A
$$\dfrac{2P_{0}T}{T+T_{0}}$$

B
$$\dfrac{P_{0}T}{T+T_{0}}$$

C
$$\dfrac{P_{0}T}{2(T+T_{0}})$$

D
$$\dfrac{T+T_{0}}{P_{0}}$$

Solution
Question 10
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A gas has molar heat capacity $$C = 4.5\ R$$ in the process $$PT = constant$$. Find the number of degrees of freedom (n) of molecules in the gas.

A
$$n = 7$$

B
$$n = 3$$

C
$$n = 5$$

D
$$n = 2$$

Solution