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Question: Two moles of an ideal gas at 2 atm and \({27^ \circ }\)C are compressed isothermally to half the vol...

Two moles of an ideal gas at 2 atm and 27∘{27^ \circ }C are compressed isothermally to half the volume by an external pressure at 4 atm. Calculate w, q and ΔU\Delta U.

Explanation

Solution

An ideal gas is a hypothetical gas composed of many randomly moving point particles. An ideal gas obeys ideal gas law. We shall calculate the internal energy from the formula given and use the 1st law of thermodynamics to calculate q and w.
Formula used: ΔU=f2nRT\Delta U = \dfrac{f}{2}nRT
ΔU=ΔQ+W\Delta U = \Delta Q + W

Complete step by step answer:
The first law of thermodynamics states that energy can be transformed from one form to another but it can neither be created nor destroyed. It is given by the formula which is as follows,
ΔU=ΔQ+W\Delta U = \Delta Q + W
where Q is the quantity of energy supplied as heat, U denotes the change in the internal energy and W denotes the amount of thermodynamic work done by the system.
A thermodynamic process in which the temperature of a system remains constant is called an isothermal process.
Internal energy is the random, disordered motion of molecules. Internal energy depends on the absolute temperature. The formula for internal energy is given as follows,
ΔU=f2nRT\Delta U = \dfrac{f}{2}nRT
In an isothermal process, the temperature remains constant and hence due to constant temperature, the change in internal energy is zero. Therefore, the value of ΔU\Delta U is equal to zero.
The formula for work done is given as follows,
W=−2.303nRTlog⁡V2V1W = - 2.303nRT\log \dfrac{{{V_2}}}{{{V_1}}}
The moles of an ideal gas compressed in this reaction are 2 and volume reduces to half.
Hence, the work done is as follows,
W=−2.303nRTlog⁡V/2V=34.66 litre atmW = - 2.303nRT\log \dfrac{{V/2}}{V} = 34.66{\text{ litre atm}}
Q=−W=−34.66 litre atmQ = - W = - 34.66{\text{ litre atm}}

Note:
Real gas doesn't obey the ideal gas equation. It obeys Vander Waals equation. Unless mentioned otherwise, we shall always assume the gas to follow ideal behaviour.The first law of thermodynamics is connected with the law of conservation of energy. The first law of thermodynamics relates heat, work and internal energy.