A monoatomic gas having $\gamma = \frac{5}{3}$ is stored in a thermally insulated container and the gas is suddenly compressed to $\frac{1}{8}^{\text{th}}$ of its initial volume. The ratio of final pressure and initial pressure is: ($\gamma$ is the ratio of specific heats of the gas at constant pressure and at constant volume)
Adiabatic Compression Pressure Ratio Calculation
The problem involves a monoatomic gas undergoing a sudden compression in a thermally insulated container. This scenario represents an adiabatic process, where no heat is exchanged between the system and the surroundings. The relationship between pressure ($P$) and volume ($V$) for an adiabatic process is given by:
$P V^\gamma = \text{constant}$
Where $\gamma$ is the adiabatic index (ratio of specific heats).
Applying the Adiabatic Formula
For the initial state (1) and the final state (2), we can write:
$P_1 V_1^\gamma = P_2 V_2^\gamma$
We need to find the ratio of the final pressure ($P_2$) to the initial pressure ($P_1$). Rearranging the formula:
$\frac{P_2}{P_1} = \left(\frac{V_1}{V_2}\right)^\gamma$
Step-by-Step Calculation
- Identify Given Values:
- Adiabatic index, $\gamma = \frac{5}{3}$.
- The gas is compressed to $\frac{1}{8}^{\text{th}}$ of its initial volume, so $\frac{V_2}{V_1} = \frac{1}{8}$. This implies $\frac{V_1}{V_2} = 8$.
- Substitute Values into the Formula:
Substitute the values of $\frac{V_1}{V_2}$ and $\gamma$ into the pressure ratio equation:
$\frac{P_2}{P_1} = (8)^{\frac{5}{3}}$
- Simplify the Expression:
Express 8 as a power of 2 ($8 = 2^3$):
$\frac{P_2}{P_1} = (2^3)^{\frac{5}{3}}$
Using the power rule $(a^m)^n = a^{m \times n}$:
$\frac{P_2}{P_1} = 2^{3 \times \frac{5}{3}} = 2^5$
- Final Result:
Calculate $2^5$:
$2^5 = 32$
Therefore, the ratio of the final pressure and initial pressure ($\frac{P_2}{P_1}$) is 32.
Similar Questions
During the melting of a slab of ice at $273 \ K$ at atmospheric pressure:
- $\gamma_A$ is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. $\gamma_B$ is the specific heat ratio of polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If $\frac{\gamma_A}{\gamma_B} = \left(1 + \frac{1}{n}\right)$, then the value of n is _________.
- Match List - I with List - II.$\Delta Q$=Heat supplied
List - I List - II (A) Isobaric (I) $\Delta Q = \Delta W$ (B) Isochoric (II) $\Delta Q = \Delta U$ (C) Adiabatic (III) $\Delta Q = \text{zero}$ (D) Isothermal (IV) $\Delta Q = \Delta U + P\Delta V$
$\Delta W$ = Work done by the system
$\Delta U$=Change in internal energy
P = Pressure of the system
$\Delta V$ = Change in volume of the system
Choose the correct answer from the options given below : - There are two vessels filled with an ideal gas where volume of one is double the volume of other. The large vessel contains the gas at 8 kPa at 1000 K while the smaller vessel contains the gas at 7 kPa at 500 K. If the vessels are connected to each other by a thin tube allowing the gas to flow and the temperature of both vessels is maintained at 600 K, at steady state the pressure in the vessels will be (in kPa).
Match the LIST-I with LIST-II Choose the correct answer from the options given below:
An ideal gas has undergone through the cyclic process as shown in the figure. Work done by the gas in the entire cycle is ________ $\times 10^{-1}$J.
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- An ideal gas exists in a state with pressure $P_0$, volume $V_0$. It is isothermally expanded to $4$ times of its initial volume($V_0$), then isobarically compressed to its original volume. Finally the system is heated isochorically to bring it to its initial state. The amount of heat exchanged in this process is
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(Take $g = 10 \text{ m/s}^2$, specific heat of water $= 4200 \text{ J/(kg K)}$)
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Important Questions from Heat and Thermodynamics
During the melting of a slab of ice at $273 \ K$ at atmospheric pressure:
- $\gamma_A$ is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. $\gamma_B$ is the specific heat ratio of polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If $\frac{\gamma_A}{\gamma_B} = \left(1 + \frac{1}{n}\right)$, then the value of n is _________.
- Match List - I with List - II.$\Delta Q$=Heat supplied
List - I List - II (A) Isobaric (I) $\Delta Q = \Delta W$ (B) Isochoric (II) $\Delta Q = \Delta U$ (C) Adiabatic (III) $\Delta Q = \text{zero}$ (D) Isothermal (IV) $\Delta Q = \Delta U + P\Delta V$
$\Delta W$ = Work done by the system
$\Delta U$=Change in internal energy
P = Pressure of the system
$\Delta V$ = Change in volume of the system
Choose the correct answer from the options given below : - There are two vessels filled with an ideal gas where volume of one is double the volume of other. The large vessel contains the gas at 8 kPa at 1000 K while the smaller vessel contains the gas at 7 kPa at 500 K. If the vessels are connected to each other by a thin tube allowing the gas to flow and the temperature of both vessels is maintained at 600 K, at steady state the pressure in the vessels will be (in kPa).
Match the LIST-I with LIST-II Choose the correct answer from the options given below:
