(Take the universal gas constant as $R = 8.3\text{ J mol}^{-1}\text{ K}^{-1}$)
Key Formula for Adiabatic Process
For an adiabatic process involving an ideal gas, the work done ($W$) is related to the change in internal energy ($\Delta U$). According to the first law of thermodynamics, $\Delta U = Q - W$. Since the process is adiabatic, $Q = 0$, hence $\Delta U = -W$.
The change in internal energy for $n$ moles of an ideal gas is given by:
$\Delta U = n C_V \Delta T$
Where $C_V$ is the molar specific heat at constant volume and $\Delta T$ is the change in temperature. For a monatomic ideal gas, $C_V = \frac{R}{\gamma - 1}$.
Combining these, the work done by the gas is:
$W = -\Delta U = -n C_V \Delta T = -n \left(\frac{R}{\gamma - 1}\right) (T_2 - T_1) = n \left(\frac{R}{\gamma - 1}\right) (T_1 - T_2)$
Step-by-Step Calculation
- Identify the given values:
- Number of moles, $n = 1$ mol
- Adiabatic index, $\gamma = 5/3$
- Initial temperature, $T_1 = 60$ K
- Final temperature, $T_2 = 50$ K
- Universal gas constant, $R = 8.3$ J mol-1 K-1
- Calculate the term $(\gamma - 1)$:
$\gamma - 1 = \frac{5}{3} - 1 = \frac{5 - 3}{3} = \frac{2}{3}$
- Calculate the temperature difference $(T_1 - T_2)$:
$T_1 - T_2 = 60 \text{ K} - 50 \text{ K} = 10 \text{ K}$
- Substitute the values into the work done formula:
$W = n \left(\frac{R}{\gamma - 1}\right) (T_1 - T_2)$
$W = (1 \text{ mol}) \left(\frac{8.3 \text{ J mol}^{-1} \text{ K}^{-1}}{2/3}\right) (10 \text{ K})$
- Simplify the expression:
$W = 1 \times \left(8.3 \times \frac{3}{2}\right) \times 10 \text{ J}$
$W = \frac{3}{2} \times 8.3 \times 10 \text{ J}$
$W = 1.5 \times 8.3 \times 10 \text{ J}$
$W = 12.45 \times 10 \text{ J}$
$W = 124.5 \text{ J}$
- Conclusion: The work done by the gas is $124.5$ J.
Similar Questions
- An electric heater supplies heat to a system at a rate of $100 \text{ W}$. If the system performs work at a rate of $75 \text{ J/s}$, then the rate at which internal energy increases will be :
A flask contains argon and chlorine in the ratio of $2:1$ by mass. The temperature of the mixture is $27^\circ\text{C}$. The ratio of root mean square speed of the molecules of the two gases $(\frac{V_{rms}^{Ar}}{V_{rms}^{Cl}})$ is :
(Atomic mass of argon = $40 \text{ u}$ and molecular mass of chlorine = $70 \text{ u}$)- The mean free path of molecules in an ideal gas A is half that of another ideal gas B. The diameter of the spherical molecules of gas A is twice the diameter of the molecules of B. If number densities of the gases A and B are $n_A$ and $n_B$, respectively, then the correct option is :
One mole of an ideal monatomic gas undergoes a cyclic process as shown in the figure. The total heat supplied to the gas is :
- An ideal gas is made of polyatomic molecules. Each of the molecules has three translational, three rotational and $f$ number of vibrational modes. If the ratio of heat capacities $C_P/C_V$ of the gas is 8/7, then the value of $f$ is :
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.
