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Q12.3 (a):
Figure 12.8 shows plot of (\frac{PV}{T}) versus (P) for (1.00 \times 10^{-3}) kg of oxygen gas at two different temperatures. (a) What does the dotted plot signify?
Figure 12.8 shows plot of (\frac{PV}{T}) versus (P) for (1.00 \times 10^{-3}) kg of oxygen gas at two different temperatures. (a) What does the dotted plot signify?
Solution :
1. Theoretical Foundation: The Ideal Gas Law
The state of a gas is described by the Ideal Gas Equation: $PV = \mu RT$, where:
- $P$ is the pressure ($Pa$)
- $V$ is the volume ($m^3$)
- $\mu$ is the number of moles ($\mu = \frac{m}{M}$, where $m$ is mass and $M$ is molar mass)
- $R$ is the Universal Gas Constant ($8.314 \, J \cdot mol^{-1} \cdot K^{-1}$)
- $T$ is the absolute temperature ($K$)
Rearranging this equation to match the vertical axis of the provided plot, we obtain:
$\frac{PV}{T} = \mu R$ [By algebraic transposition of the Ideal Gas Law]
2. Analysis of the Ideal Gas Behavior
For a strictly ideal gas, the quantity $\frac{PV}{T}$ is a constant equal to $\mu R$. Consequently, if we plot $\frac{PV}{T}$ against $P$, an ideal gas would result in a horizontal straight line parallel to the $P$-axis, indicating that the ratio is independent of pressure variations. [Per the definition of an Ideal Gas in Kinetic Theory]
3. Interpreting the Dotted Plot
In physical reality, real gases (such as Oxygen) deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and the finite volume of gas molecules. The Van der Waals equation, $(P + \frac{an^2}{V^2})(V - nb) = nRT$, accounts for these deviations. As $P \to 0$, the behavior of any real gas converges toward the ideal gas limit. The dotted line in the figure represents this limiting case where the gas molecules are sufficiently far apart that their individual volumes and intermolecular attractions become negligible. [Per Kinetic Theory of Gases, specifically the limit of low density]
4. Logical Conclusion
The dotted plot signifies the behavior of an ideal gas. It represents the theoretical value of $\mu R$ which real gases approach as the pressure $P$ decreases towards zero. In this limit, the assumptions of the Kinetic Theory (neglecting molecular volume and inter-molecular forces) hold true, and the real gas behaves perfectly according to the Ideal Gas Law.
Final Solution: The dotted plot signifies the behavior of an ideal gas, serving as a theoretical reference point to which real oxygen gas converges as pressure approaches zero.
More Questions from Class 11 Physics Kinetic Theory EXERCISES
- Q12.1: Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3 Å.
- Q12.10: Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature (17 °\text{C}). Take the radius of a nitrogen molecule to be roughly 1.0 Å. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of (N_{2}) = 28.0 u).
- Q12.2: Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and pressure (STP : 1 atmospheric pressure, (0 °\text{C})). Show that it is 22.4 litres.
- Q12.3 (b): Figure 12.8 shows plot of (\frac{PV}{T}) versus (P) for (1.00 \times 10^{-3}) kg of oxygen gas at two different temperatures. (b) Which is true: (T_{1} > T_{2}) or (T_{1} < T_{2})?
- Q12.3 (c): Figure 12.8 shows plot of (\frac{PV}{T}) versus (P) for (1.00 \times 10^{-3}) kg of oxygen gas at two different temperatures. (c) What is the value of (\frac{PV}{T}) where the curves meet on the y-axis?
- Q12.3 (d): Figure 12.8 shows plot of (\frac{PV}{T}) versus (P) for (1.00 \times 10^{-3}) kg of oxygen gas at two different temperatures. (d) If we obtained similar plots for (1.00 \times 10^{-3}) kg of hydrogen, would we get the same value of (\frac{PV}{T}) at the point where the curves meet on the y-axis? If not, what mass of hydrogen yields the same value of (\frac{PV}{T}) (for low pressure high temperature region of the plot) ? (Molecular mass of (H_{2}) = 2.02 u, of (O_{2}) = 32.0 u, (R) = 8.31 J (\text{mol}^{-1}) (\text{K}^{-1}).)
- Q12.4: An oxygen cylinder of volume 30 litre has an initial gauge pressure of 15 atm and a temperature of (27 °\text{C}). After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to (17 °\text{C}). Estimate the mass of oxygen taken out of the cylinder ((R) = 8.31 J (\text{mol}^{-1}) (\text{K}^{-1}), molecular mass of (O_{2}) = 32 u).
- Q12.5: An air bubble of volume (1.0 \text{ cm}^{3}) rises from the bottom of a lake 40 m deep at a temperature of (12 °\text{C}). To what volume does it grow when it reaches the surface, which is at a temperature of (35 °\text{C}) ?
- Q12.6: Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity (25.0 \text{ m}^{3}) at a temperature of (27 °\text{C}) and 1 atm pressure.
- Q12.7 (i): Estimate the average thermal energy of a helium atom at (i) room temperature ((27 °\text{C})),
- Q12.7 (ii): Estimate the average thermal energy of a helium atom at (ii) the temperature on the surface of the Sun (6000 K),
- Q12.7 (iii): Estimate the average thermal energy of a helium atom at (iii) the temperature of 10 million kelvin (the typical core temperature in the case of a star).
- Q12.8: Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic), the second contains chlorine (diatomic), and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules ? Is the root mean square speed of molecules the same in the three cases? If not, in which case is (v_{rms}) the largest ?
- Q12.9: At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at (– 20 °\text{C}) ? (atomic mass of Ar = 39.9 u, of He = 4.0 u).
- Q13.1: Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3 Å.
- Q13.10: Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature 17 0C. Take the radius of a nitrogen molecule to be roughly 1.0 Å. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of $N_2$ = 28.0 u).
- Q13.2: Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and pressure (STP : 1 atmospheric pressure, 0 °C). Show that it is 22.4 litres.
- Q13.3(a): Figure 13.8 shows plot of $frac{PV}{T}$ versus $P$ for $1.00 \times 10^{-3}$ kg of oxygen gas at two different temperatures. (a) What does the dotted plot signify?
- Q13.3(b): Figure 13.8 shows plot of $frac{PV}{T}$ versus $P$ for $1.00 \times 10^{-3}$ kg of oxygen gas at two different temperatures. (b) Which is true: $T_1 > T_2$ or $T_1 < T_2$?
- Q13.3(c): Figure 13.8 shows plot of $frac{PV}{T}$ versus $P$ for $1.00 \times 10^{-3}$ kg of oxygen gas at two different temperatures. (c) What is the value of $frac{PV}{T}$ where the curves meet on the y-axis?
- Q13.3(d): Figure 13.8 shows plot of $frac{PV}{T}$ versus $P$ for $1.00 \times 10^{-3}$ kg of oxygen gas at two different temperatures. (d) If we obtained similar plots for $1.00 \times 10^{-3}$ kg of hydrogen, would we get the same value of $frac{PV}{T}$ at the point where the curves meet on the y-axis? If not, what mass of hydrogen yields the same value of $frac{PV}{T}$ (for low pressure high temperature region of the plot) ? (Molecular mass of $H_2$ = 2.02 u, of $O_2$ = 32.0 u, $R = 8.31$ J $mol^{-1} K^{-1}$.)
- Q13.4: An oxygen cylinder of volume 30 litres has an initial gauge pressure of 15 atm and a temperature of 27 °C. After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to 17 °C. Estimate the mass of oxygen taken out of the cylinder ($R = 8.31$ J $mol^{-1} K^{-1}$, molecular mass of $O_2$ = 32 u).
- Q13.5: An air bubble of volume 1.0 $cm^3$ rises from the bottom of a lake 40 m deep at a temperature of 12 °C. To what volume does it grow when it reaches the surface, which is at a temperature of 35 °C ?
- Q13.6: Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 $m^3$ at a temperature of 27 °C and 1 atm pressure.
- Q13.7(i): Estimate the average thermal energy of a helium atom at (i) room temperature (27 °C),
- Q13.7(ii): Estimate the average thermal energy of a helium atom at (ii) the temperature on the surface of the Sun (6000 K),
- Q13.7(iii): Estimate the average thermal energy of a helium atom at (iii) the temperature of 10 million kelvin (the typical core temperature in the case of a star).
- Q13.8: Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic), the second contains chlorine (diatomic), and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules ? Is the root mean square speed of molecules the same in the three cases? If not, in which case is $v_{rms}$ the largest ?
- Q13.9: At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at – 20 °C ? (atomic mass of $Ar$ = 39.9 u, of $He$ = 4.0 u).
CBSE Solutions for Class 11 Physics Kinetic Theory
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