A gas is kept in a container having walls which are thermally non-conducting. Initially the gas has a volume of 800 $ cm^3 $ and temperature 27°C. The change in temperature when the gas is adiabatically compressed to 200 $ cm^3 $ is: (Take $ \gamma $ = 1.5 : $ \gamma $ is the ratio of specific heats at constant pressure and at constant volume)
Show Hint
- 327 K
- 600 K
- 522 K
- 300 K
The Correct Option is D
Approach Solution - 1
The problem asks for the change in temperature of a gas that is adiabatically compressed from an initial volume and temperature to a final volume. The container has thermally non-conducting walls, which confirms the process is adiabatic.
Concept Used:
For a reversible adiabatic process, the relationship between the absolute temperature (T) and volume (V) of a gas is given by the equation:
\[ TV^{\gamma-1} = \text{constant} \]where \( \gamma \) is the ratio of specific heats (\( C_p/C_v \)). For two states, initial (1) and final (2), this relationship can be written as:
\[ T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1} \]Step-by-Step Solution:
Step 1: List the given initial and final state variables and convert the temperature to Kelvin.
Given values are:
- Initial Volume, \( V_1 = 800 \, \text{cm}^3 \)
- Final Volume, \( V_2 = 200 \, \text{cm}^3 \)
- Ratio of specific heats, \( \gamma = 1.5 \)
- Initial Temperature, \( T_1 = 27^\circ \text{C} \)
In thermodynamics, we must use the absolute temperature scale (Kelvin). The conversion is \( T(\text{K}) = T(^\circ\text{C}) + 273 \).
\[ T_1 = 27 + 273 = 300 \, \text{K} \]Step 2: Set up the adiabatic relation to find the final temperature \( T_2 \).
Using the relation \( T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1} \), we can solve for \( T_2 \):
\[ T_2 = T_1 \left( \frac{V_1}{V_2} \right)^{\gamma-1} \]Step 3: Substitute the given values into the equation and calculate \( T_2 \).
First, calculate the exponent \( \gamma - 1 \):
\[ \gamma - 1 = 1.5 - 1 = 0.5 = \frac{1}{2} \]Next, calculate the ratio of the volumes:
\[ \frac{V_1}{V_2} = \frac{800 \, \text{cm}^3}{200 \, \text{cm}^3} = 4 \]Now, substitute these values into the equation for \( T_2 \):
\[ T_2 = 300 \, \text{K} \times (4)^{0.5} = 300 \, \text{K} \times \sqrt{4} \] \[ T_2 = 300 \, \text{K} \times 2 = 600 \, \text{K} \]Final Computation & Result:
The question asks for the change in temperature, which is \( \Delta T = T_2 - T_1 \).
\[ \Delta T = 600 \, \text{K} - 300 \, \text{K} = 300 \, \text{K} \]The change in temperature is 300 K. This corresponds to option (4).
Approach Solution -2
\( V_1 = 800 cm^3 \) \( V_2 = 200 cm^3 \) \( T_1 = 300 \)
K For adiabatic: \( TV^{\gamma - 1} = \) constant.
\( T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \) \( (300) (800)^{1.5 - 1} = T_2 (200)^{1.5 - 1} \) \( T_2 = 300 \left( \frac{800}{200} \right)^{0.5} \) \( T_2 = 300 (4)^{1/2} \)
\( T_2 = 600 \) K \( \Delta T = 600 - 300 = 300 \) K
Top JEE Main Physics Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
- $I _{ CM }$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc $I_{A B}$ is it's moment of inertia about an axis $AB$ perpendicular to plane and parallel to axis $CM$ at a distance $\frac{2}{3} R$ from center Where $R$ is the radius of the dise The ratio of $I _{ AB }$ and $I _{ CM }$ is $x: 9$ The value of $x$ is______
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
Top JEE Main Thermodynamics Questions
- N moles of a polyatomic gas (f = 6) must be mixed with two moles of a monoatomic gas so that the mixture behaves as a diatomic gas. The value of N is :
- 0.08 kg of air is heated at constant volume through 5°C. The specific heat of air at constant volume is 0.17 kcal/kg°C and \( J = 4.18 \) joule/cal. The change in its internal energy is approximately:
- The given figure represents two isobaric processes for the same mass of an ideal gas, then
- A real gas within a closed chamber at \( 27^\circ \text{C} \) undergoes the cyclic process as shown in the figure. The gas obeys \( PV^3 = RT \) equation for the path \( A \) to \( B \). The net work done in the complete cycle is (assuming \( R = 8 \, \text{J/mol K} \)):
- At room temperature \( 27^\circ\text{C} ,\) the resistance of a heating element is \(50 \, \Omega\). The temperature coefficient of the material is \(2.4 \times 10^{-4}\) \(^\circ\text{C}^{-1} \). The temperature of the element, when its resistance is \(62 \, \Omega\) , is _________\(\degree C\).
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.