Two moles of an ideal monoatomic gas is confined in a...

Two moles of an ideal monoatomic gas is confined in a cylinder by a spring loaded piston of cross-section area $4 \times 10^{-3} \mathrm{~m}$ - Initially the spring of spring constant $k=1920 \mathrm{~N} / \mathrm{m}$ is in its relaxed state. Now, the gas was heated by an electric heater, placed inside the cylinder, for some time and due to which gas expands and does $50 \mathrm{~J}$ of work in moving piston through a distance of $0.1 \mathrm{~m}$. The temperature of the gas increases by $50 \mathrm{~K}$. Assume piston and spring to be massless and there is no friction between the piston and the cylinder. Heat supplied by the heater: (a) $1295 \mathrm{~J}$ (b) $1200 \mathrm{~J}$ (c) $1195 \mathrm{~J}$ (d) $1350 \mathrm{~J}$

Two moles of an ideal monoatomic gas is confined in a cylinder by a spring loaded piston of cross-section area $4 \times 10^{-3} \mathrm{~m}$ - Initially the spring of spring constant $k=1920 \mathrm{~N} / \mathrm{m}$ is in its relaxed state. Now, the gas was heated by an electric heater, placed inside the cylinder, for some time and due to which gas expands and does $50 \mathrm{~J}$ of work in moving piston through a distance of $0.1 \mathrm{~m}$. The temperature of the gas increases by $50 \mathrm{~K}$. Assume piston and spring to be massless and there is no friction between the piston and the cylinder. Heat supplied by the heater:
(a) $1295 \mathrm{~J}$
(b) $1200 \mathrm{~J}$
(c) $1195 \mathrm{~J}$
(d) $1350 \mathrm{~J}$

Key Concepts

Spring Potential Energy

Spring potential energy refers to the energy stored in a spring when it is compressed or stretched. In systems where a spring-loaded piston is present, the work done by the gas not only results in mechanical displacement but also contributes to the elastic potential energy stored in the spring, which must be accounted for in the overall energy balance.

Internal Energy of an Ideal Gas

For an ideal gas, the internal energy is a function solely of its temperature. In monoatomic gases, the internal energy change is determined by the temperature change and can be computed using degrees-of-freedom considerations. This concept is critical when calculating how heating affects the energy stored within the gas, independent of the mechanical work being performed.

Work in Thermodynamic Processes

Work in thermodynamic processes involves energy transfer when a system expands or compresses against an external force. In the context of gas expansion, the work done by the gas is an integral part of the energy balance, as it contributes to the net energy change of the system alongside changes in internal energy.

Ideal Gas Law

The ideal gas law relates the pressure, volume, and temperature of a gas under the assumption that there are no intermolecular forces and that the molecules occupy negligible volume. This law provides a baseline for predicting the behavior of gases in various conditions and is fundamental in connecting macroscopic measurements with molecular behavior, which is crucial in analyzing gas processes such as expansion and heating.

First Law of Thermodynamics

The First Law of Thermodynamics states that energy is conserved in any process, meaning that the heat added to a system equals the increase in its internal energy plus the work done by the system. This principle is central to thermodynamic analyses because it allows one to account for all energy transfers and conversions in processes involving heating, expansion, or compression.

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