Ideal Gas Equation

Q.

A sample of an ideal gas has volume V, pressure P and temperature T. The mass of each molecule of the gas is m. The density of the gas is (k is the Boltzmann's constant )

1. mkT

2. $\frac{P}{kT}$

3. $\frac{P}{kVT}$

4. $\frac{mP}{kT}$

Q. A graph is plotted with $PV/T$ on $y-$ axis and mass of the gas along $x-$ axis for different gases. The graph is

1. a straight line parallel to $x-$ axis for all the gases.
2. a straight line passing through origin with a slope having a constant value for all the gases.
3. a straight line passing through origin with a slope having different values for different gases.
4. a straight line parallel to $y-$ axis for all the gases.

Q. Two vessels separately contains two ideal gases $A$ and $B$ at the same temperature. The pressure of $A$ being twice that of $B$. Under such conditions, the density of $A$ is found to be $1.5$ times the density of $B$. The ratio of molecular weight of $A$ and $B$ is

1. $1:2$
2. $2:3$
3. $3:4$
4. $4:5$

Q. $n$ moles of a gas filled in a container at temperature $T$ is in thermodynamic equilibrium initially. If the gas is compressed slowly and isothermally to half its initial volume then the work done by the atmosphere on the piston is -

1. $nRT\ln 2$
2. $\frac{nRT}{2}$
3. $-\frac{3nRT}{2}$
4. $\frac{3nRT\ln 2}{2}$

Q. One mole of an ideal gas goes through the process $P=\frac{2V^2}{1+V^2}$. Then, the change in temperature (in $\text{K}$) of the gas when volume changes from $1{\text{m}}^3$ to $2{\text{m}}^3$ is

1. $\frac{4}{5R}$
2. $\frac{11}{5R}$
3. $\frac{5}{2R}$
4. $\frac{5}{3R}$

Q. An ideal monoatomic gas at temperature ${27}^{\circ }\text{C}$ and pressure $100\text{bar}$ occupies $5\text{L}$ volume. $5000\text{cal}$ of heat is added to the system without changing the volume. Calculate the change in temperature of the gas. (Given, $R=8.31\text{J/mol K}$)

1. $75\text{K}$
2. $89\text{K}$
3. $83\text{K}$
4. $90\text{K}$

Q. Figure (24 - E3) shows two vessels A and B with rigid walls containing ideal gases. The pressure, temperature and the volume are $p_A,T_A,V$ in the vessel A and $p_B,T_B,V$ in the vessel B. The vessels are now connected through a small tube. The ratio between P and T is

1. $\frac{1}{2}(\frac{P_A}{T_A}+\frac{P_B}{T_B})$
2. $\frac{1}{2}(\frac{P_A}{T_A}+\frac{P_B}{T_B})R$
3. $(\frac{P_A}{T_B}+\frac{P_B}{T_A})$
4. None

Q. One litre of oxygen at a pressure of $1\text{atm}$ and two litres of nitrogen at a pressure of $0.5\text{atm}$ at the same temperature are introduced into a vessel of one litre volume. If there is no change in the temperature, then final pressure of the mixture of gases is

1. $1\text{atm}$
2. $3\text{atm}$
3. $2\text{atm}$
4. $4\text{atm}$

Q. An ideal monoatomic gas is confined in a cylinder by a spring-loaded piston of cross-sectional area $8\times {10}^{-3}{\text{m}}^2$. Initially the gas is at $300\text{K}$ and occupies a volume of $2.4\times {10}^{-3}{\text{m}}^3$. The spring is in a relaxed state. The gas is heated by a small heater coil $H$. The force constant of the spring is $8000\text{N/m}$ and the atmospheric pressure is $1.0\times {10}^5\text{Pa}$. The cylinder and piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and cylinder. There is no heat loss through the heater coil wire and thermal capacity of the heater coil is negligible. With all the above assumptions, if the gas is heated by the heater until the piston moves out slowly by $0.1\text{m}$, then the final temperature is -

1. $400\text{K}$
2. $800\text{K}$
3. $1200\text{K}$
4. $300\text{K}$

Q. A balloon contains $1500{\text{m}}^3$ of helium gas at $27{}^{\circ }\text{C}$ and $4\text{atm}$ . The volume of helium gas at $-3{}^{\circ }\text{C}$ and $2\text{atm}$ will be

1. $2700{\text{m}}^3$
2. $1900{\text{m}}^3$
3. $1700{\text{m}}^3$
4. $1500{\text{m}}^3$

Q. An ideal monoatomic gas at temperature ${27}^{\circ }\text{C}$ and pressure $100\text{bar}$ occupies $5\text{L}$ volume. $5000\text{cal}$ of heat is added to the system without changing the volume. Calculate the change in temperature of the gas. (Given, $R=8.31\text{J/mol K}$)

1. $75\text{K}$
2. $89\text{K}$
3. $83\text{K}$
4. $90\text{K}$

Q. A certain amount of an ideal gas is initially at a pressure $P_1$ and temperature $T_1$. First, it undergoes an isobaric process 1 - 2 such that $T_2=\frac{3T_1}{4}$. Then, it undergoes an isochoric process $2\to 3$ such that $T_3=\frac{T_2}{2}$. The ratio of the final volume to the initial volume of the ideal gas is

1. $0.25$
2. $1$
3. $1.5$
4. $0.75$

Q. $12\text{g}$ $\text{He}$ and $4\text{g}{\text{H}}_2$ filled in a container of volume $20\text{L}$ maintained at temperature $300\text{K}$. The pressure of the mixture is nearly

1. $3.225\text{atm}$
2. $5.225\text{atm}$
3. $6.225\text{atm}$
4. $1.225\text{atm}$

Q. A certain amount of an ideal gas is initially at a pressure $P_1$ and temperature $T_1$. First, it undergoes an isobaric process 1 - 2 such that $T_2=\frac{3T_1}{4}$. Then, it undergoes an isochoric process $2\to 3$ such that $T_3=\frac{T_2}{2}$. The ratio of the final volume to the initial volume of the ideal gas is

1. $0.25$
2. $1$
3. $1.5$
4. $0.75$

Q.

A closed hollow insulated cylinder is filled with gas at $0^{o}C$ and also contains an insulated piston of negligible weight and negligible thickness at the middle point. The gas on one side is heated to ${100}^{o}C$. If the piston moves through 5 cm, the length of the hollow cylinder is

1. 13.65 cm

2. 27.3 cm

3. 38.6 cm

4. 64.6 cm

Q. The air density at Mount Everest is less than that at the sea level. Mountaineers find that for one trip lasting a few hours, the extra oxygen needed by them corresponds to $30,000\text{cc}$ at sea level (pressure 1 atmosphere, temperature ${27}^{\circ }\text{C}$). Assuming that the temperature around Mount Everest is $–{73}^{\circ }\text{C}$ and that the oxygen cylinder has a capacity of $5.2\text{litre}$, the pressure at which $O_2$ be filled (at site) in the cylinder is

1. $3.86\text{atm}$
2. $5.00\text{atm}$
3. $5.77\text{atm}$
4. $1\text{atm}$

Q. Figure 24-Q1 shows graphs of pressure vs density for an ideal gas at two temperatures $T_1$ and $T_2$.

1. $T_1>T_2$
2. $T_1=T_2$
3. $T_1<T_2$
4. Any of the three is possible.

Q.

The equation of state of a gas is $(P+\frac{a{T}^2}{V})\times V_1^c=(RT+b)$ where a, b, c and R are constants. The isotherms can be represented by $P=A{V}^m-B{V}^n$ where A and B depend only on temperature. Then,

1. m = - c, n = -1

2. m = c, n = 1

3. m = -c, n = 1

4. m = c, n = -1

Q. Two vessels A and B having equal volume contain equal masses of hydrogen in A and helium in B at $300\text{K}$. Then, mark the correct statement.

1. The pressure exerted by hydrogen is half that of exerted by helium.
2. The pressure exerted by hydrogen is equal to that of exerted by helium.
3. The pressure exerted by helium is thrice that of exerted by hydrogen.
4. The pressure exerted by hydrogen is twice that of exerted by helium.

Q. A vessel has $6\text{g}$ of hydrogen at pressure $P$ and temperature $500\text{K}$. A small hole is made in it so that hydrogen leaks out. How much hydrogen leaks out if the final pressure is $P/2$ and temperature falls to $300\text{K}$ when the hole is sealed?

1. $2\text{g}$
2. $3\text{g}$
3. $4\text{g}$
4. $1\text{g}$

Q. For an ideal gas the fractional change in its volume per degree rise in temperature at constant pressure is equal to
$[T$ is the absolute temperature of gas $]$

1. $T^2$
2. $T^1$
3. $T^{-1}$
4. $T^{-2}$

Q.

Two thermally insulated vessels 1 and 2 are filled with air at temperatures $(T_1,T_2)$, volume $(V_1,V_2)$ and pressure $(P_1,P_2)$ respectively. If the valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be

1. $T_1+T_2$
   

2. $(T_1+T_2)/2$
   

3. $T_1{T}_2(P_1{V}_1+P_2{V}_2)/(P_1{V}_1{T}_2+P_2{V}_2{T}_1)$
   

4. $T_1{T}_2(P_1{V}_1+P_2{V}_2)/(P_1{V}_1{T}_1+P_2{V}_2{T}_2)$

Q. Two balloons are filled, one with pure $He$ gas and other by air. If the pressure and temperature of these balloons are same, then the number of molecules per unit volume is:

1. more in the $He$ filled balloon.
2. same in both the balloons.
3. more in the air filled balloon.
4. in the ratio of $1:4$.

Q.

A closed hollow insulated cylinder is filled with gas at $0^{o}C$ and also contains an insulated piston of negligible weight and negligible thickness at the middle point. The gas on one side is heated to ${100}^{o}C$. If the piston moves through 5 cm, the length of the hollow cylinder is

1. 13.65 cm

2. 27.3 cm

3. 38.6 cm

4. 64.6 cm

Q. Two vessels separately contain two ideal gases $A$ and $B$ at the same temperature. The pressure of $A$ is twice that of $B$. Under such conditions, the density of $A$ is found to be 1.5 times the density of $B$. The ratio of molecular weight of $A$ and $B$ is

1. $\frac{1}{2}$
2. $\frac{2}{3}$
3. $\frac{3}{4}$
4. $2$

Q.

An enclosure of volume V contains a mixture of 8g of oxygen, 14g of nitrogen and 22g of carbon dioxide at absolute temperature T. The pressure of the mixture of gases is (R is universal gas constant)

1. $\frac{RT}{V}$
   

2. $\frac{3RT}{2V}$
   

3. $\frac{5RT}{4V}$
   

4. $\frac{7RT}{5V}$

Q. An ideal monoatomic gas is confined in a cylinder by a spring-loaded piston of cross-sectional area $8\times {10}^{-3}{\text{m}}^2$. Initially the gas is at $300\text{K}$ and occupies a volume of $2.4\times {10}^{-3}{\text{m}}^3$. The spring is in a relaxed state. The gas is heated by a small heater coil $H$. The force constant of the spring is $8000\text{N/m}$ and the atmospheric pressure is $1.0\times {10}^5\text{Pa}$. The cylinder and piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and cylinder. There is no heat loss through the heater coil wire and thermal capacity of the heater coil is negligible. With all the above assumptions, if the gas is heated by the heater until the piston moves out slowly by $0.1\text{m}$, then the final temperature is -

1. $400\text{K}$
2. $800\text{K}$
3. $1200\text{K}$
4. $300\text{K}$

Q. During an experiment, an ideal gas is found to obey a law $V{P}^2=$ constant. The gas is initially at temperature $T$ and volume $V$. When it expands to volume $2V$, the resulting temperature will be

1. $\sqrt{5}T$
2. $\sqrt{3}T$
3. $\sqrt{2}T$
4. $\sqrt{6}T$