Problem Analysis
We are given the following information for an ideal gas:
- Heat supplied, $Q = 300 \text{ J}$
- Specific heat at constant pressure, $C_p = \frac{7}{2} R$
- Initial temperature, $T_1 = 20^\circ\text{C}$
- Final temperature, $T_2 = 50^\circ\text{C}$
- Process: Constant volume (isochoric)
- Universal gas constant, $R = 8.314 \text{ J/mol.K}$
We need to find the approximate mass ($m$) of the gas in grams.
Calculating Specific Heat at Constant Volume ($C_v$)
For an ideal gas, the relationship between specific heats is given by Mayer's relation: $C_p - C_v = R$
Using the given $C_p = \frac{7}{2} R$: $\frac{7}{2} R - C_v = R$ $C_v = \frac{7}{2} R - R = \frac{5}{2} R$
Substituting the value of $R$: $C_v = \frac{5}{2} \times 8.314 \text{ J/mol.K} \approx 20.785 \text{ J/mol.K}$
Calculating Temperature Change ($\Delta T$)
The change in temperature is: $\Delta T = T_2 - T_1 = 50^\circ\text{C} - 20^\circ\text{C} = 30^\circ\text{C}$
Since the difference is the same in Celsius and Kelvin, $\Delta T = 30 \text{ K}$.
Calculating Number of Moles ($n$)
For a constant volume (isochoric) process, the heat absorbed is given by: $Q = n C_v \Delta T$
Substituting the known values: $300 \text{ J} = n \times (20.785 \text{ J/mol.K}) \times (30 \text{ K})$ $300 = n \times 623.55$ $n = \frac{300}{623.55} \approx 0.481 \text{ mol}$
Calculating Gas Mass ($m$)
The value $C_p = \frac{7}{2} R$ is characteristic of a diatomic ideal gas (considering translational and rotational modes). Common diatomic gases like Nitrogen ($N_2$) have a molar mass ($M$) of approximately $28$ g/mol.
The mass of the gas is calculated using the number of moles ($n$) and the molar mass ($M$): $m = n \times M$
Assuming the gas is diatomic with $M \approx 28$ g/mol: $m \approx 0.481 \text{ mol} \times 28 \text{ g/mol}$ $m \approx 13.47 \text{ g}$
This value is approximately 13 g, consistent with the provided answer range.
Similar Questions
- Which of the following best represents the temperature versus heat supplied graph for water, in the range of $-20^\circ\text{C}$ to $120^\circ\text{C}$ ?
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Important Questions from Heat and Thermodynamics
- Which of the following best represents the temperature versus heat supplied graph for water, in the range of $-20^\circ\text{C}$ to $120^\circ\text{C}$ ?
- $10 \text{ kg}$ of ice at $-10^\circ\text{C}$ is added to $100 \text{ kg}$ of water to lower its temperature from $25^\circ\text{C}$. Consider no heat exchange to surroundings. The decrement to the temperature of water is ________$^\circ\text{C}$.
(specific heat of ice = $2100 \text{ J/Kg.}^\circ\text{C}$, specific heat of water = $4200 \text{ J/Kg.}^\circ\text{C}$, latent heat of fusion of ice = $3.36 \times 10^5 \text{ J/Kg}$) - The volume of an ideal gas increases 8 times and temperature becomes $(1/4)^{\text{th}}$ of initial temperature during a reversible change. If there is no exchange of heat in this process ($\Delta Q = 0$) then identify the gas from the following options (Assuming the gases given in the options are ideal gases):
Rods x and y of equal dimensions but of different materials are joined as shown in figure. Temperatures of end points $A$ and $F$ are maintained at $100^\circ\text{C}$ and $40^\circ\text{C}$ respectively. Given the thermal conductivity of rod x is three times of that of rod y, the temperature at junction points $B$ and $E$ are (close to):
- Consider two boxes containing ideal gases A and B such that their temperatures, pressures and number densities are same. The molecular size of A is half of that of B and mass of molecule A is four times that of B. If the collision frequency in gas B is $32 \times 10^{18}$ /s then collision frequency in gas A is _________ /s.
