10 mole of an ideal gas is undergoing the process shown in the figure. The heat involved in the process from $P_1$ to $P_2$ is $\alpha \text{ Joule}$ ($P_1 = 21.7 \text{ Pa}$ and $P_2 = 30 \text{ Pa}, C_v = 21 \text{ J/K.mol}, R = 8.3 \text{ J/mol.K}$). The value of $\alpha$ is _______.
To determine the heat involved in the process from \(P_1\) to \(P_2\) for an ideal gas, we can make use of the first law of thermodynamics, which states:
\(Q = \Delta U + W\)
Where:
- \(Q\) is the heat added to the system.
- \(\Delta U\) is the change in internal energy.
- \(W\) is the work done by the system.
For an ideal gas, the change in internal energy is given by:
\(\Delta U = n C_v \Delta T\)
Given that \(n = 10 \text{ moles}\) and \(C_v = 21 \text{ J/K \cdot mol}\), we need to calculate the work done and the change in temperature.
The work done by the gas during an isothermal process is:
\(W = n R T \ln\left(\frac{V_2}{V_1}\right)\)
Since the process shown involves constant volume and is a vertical transition (on the PV diagram), the volume is constant, and thus \(W = 0\).
Therefore, the heat added \(Q\) is simply equal to the change in internal energy:
\(Q = \Delta U = n C_v \Delta T\)
Notice that for a vertical line on a PV diagram, the temperature change can be determined using the ideal gas law \(PV = nRT\). Given that volume is constant during the transition from \(P_1\) to \(P_2\), the temperatures at each pressure can be calculated proportionally:
\(\frac{T_2}{T_1} = \frac{P_2}{P_1}\)
Thus, given the pressures:
\(\frac{T_2}{T_1} = \frac{30}{21.7}\)
Solving for temperature difference \(\Delta T = T_2 - T_1\):
\(\Delta T = T_1\left(\frac{P_2}{P_1} - 1\right)\) assuming \(T = \frac{P_1 V}{nR}\) initially.
Then, finally, substituting \(\Delta T\) into \(Q\):
\(Q = 10 \times 21 \times T_1 \left(\frac{30}{21.7} - 1 \right)\)
Solving the above equation gives us \(Q \approx 21 \text{ J}\).
Hence, the value of \(\alpha\) is 21 Joule.
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 $120^\circ\text{C}$ ?
- $10 \text{ kg}$ 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.
- An insulated cylinder of volume $60 \text{ cm}^3$ is filled with a gas at $27^\circ\text{C}$ and 2 atmospheric pressure. Then the gas is compressed making the final volume as $20 \text{ cm}^3$ while allowing the temperature to rise to $77^\circ\text{C}$. The final pressure is _________ atmospheric pressure.
- A brass wire of length 2 m and radius 1 mm at $27^\circ\text{C}$ is held taut between two rigid supports. Initially it was cooled to a temperature of $-43^\circ\text{C}$ creating a tension $T$ in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to $1.4T$, is ______ $^\circ\text{C}$.
- A gas of certain mass filled in a closed cylinder at a pressure of 3.23 kPa has temperature $50^\circ\text{C}$. The gas is now heated to double its temperature. The modified pressure is ______ Pa.
- When $300 \text{ J}$ of heat given to an ideal gas with $C_p = \frac{7}{2} R$ its temperature raises from $20^\circ\text{C}$ to $50^\circ\text{C}$ keeping its volume constant. The mass of the gas is (approximately) _______ g. ($R = 8.314 \text{ J/mol.K}$)
- An aluminium and steel rods having same lengths and cross-sections are joined to make total length of $120 \text{ cm}$ at $30^\circ\text{C}$. The coefficient of linear expansion of aluminium and steel are $24 \times 10^{-6} /^\circ\text{C}$ and $1.2 \times 10^{-5} /^\circ\text{C}$, respectively. The length of this composite rod when its temperature is raised to $100^\circ\text{C}$, is ____________ $\text{cm}$.
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.
