# Heat and Thermodynamics: Change in internal energy.

A cylinder (cross section is 0.2m2) with a free moving piston is filled with gas. The piston is attached to a heavy weight W = 10000N. Outside the cylinder, the air is at 300K and 1 atm. Initially the gas is at 300K, then it is heated to 400K. The heat capacity of the gas under the constant pressure is 500J/K.

If the length of the gas in the cylinder l increases by 20cm during the heating, find the change in the internal energy of the gas in Joule J.

(change in)U =

Please see the attachment.

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Please see the attachment.

A cylinder (cross section is 0.2m2) with a free moving piston is filled with gas. The piston is attached to a heavy weight W = 10000N. Outside the cylinder, the air is at 300K and 1 atm. Initially the gas is at 300K, then it is heated to 400K. The heat capacity of the gas under the constant pressure is 500J/K.

If the length of the gas in the cylinder l increases by 20cm during the heating, find the change in the internal energy of the gas in Joule J.

Heat remains in a gas in form of kinetic energy of the molecules of a gas and this total kinetic energy of an ideal gas is called the internal energy of the gas. As we know that the total kinetic energy of a gas depends only on the temperature of the gas, and is given by n Cv T, to find the increase in internal energy of an ideal gas we have to find only the increase in its temperature. Thus the increase in internal energy of n moles of a gas when its temperature increases by T is given by

------------ (1)

Where Cv is specific heat of the gas at constant volume and T is increase in its absolute temperature.

Let the pressure of the gas (constant) is P and the pressure of the atmosphere is Pa. the tension in the string connected to the piston is equal to the weight of the load W. hence considering the equilibrium of the piston (zero acceleration) we get the net force on the piston

Or ------------------------------ (2)

Now initial temperature of the gas T1 = 300 K

Final temperature of the gas T2 = 400 K

Initial volume of the gas V1 = l0*A (l0 is initial length)

Final volume of the gas V2 = (l0*+ l)*A (l is increase in length)

Applying Charles law at constant pressure we get

Or

Gives

Or ---------------------------------- (3)

Now the number of moles in the gas is given by the gas equation as

Hence considering the initial values and from equations (2) and (3) we get

Or ----------------------------------- (4)

Now using Mayer's equation the two specific heats of a gas are related as

Gives -------------------------------- (5)

Hence the increase in the internal energy of the gas is given by substituting values from equation (4) and (5) is equation (1) as

Or

Or

Or (C is the heat capacity of the gas)

Or

Or

Or

Thus the increase in internal energy of the gas will be 48000 J

The increase in internal energy of n mole of a gas can also be given as

DU = heat given to the gas - work done by the gas

or

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