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An ideal gas has a specific heat at cons...

An ideal gas has a specific heat at constant pressure `C_P`=(5 R/2). The gas is kept in a closed vessel of volume 0.0083 `m^3`, at a temperature of 300 K and a pressure of `1.6 xx10^6 N//m^2`. An amount of `2.49xx10^4` J of heat energy is supplied to the gas. Calculate the final temperature and pressure of the gas.

Text Solution

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Given that initial pressure, volume and temperature of gas is
`P_1=1.6xx10^6 N//m^2`
`V_1=0.0083m^3`
`T_1`=300 K
From gas law we can find the number of moles of gas in the container as
`n=(PV)/(RT)=(1.6xx10^6xx0.0083)/(8.314xx300)`
`=16/3`mole
It is given that molar specific heat of gas at constant pressure is
`C_P=(5R)/2`
Thus gas is monoatomic, hence its molar specific heat at constant volume is given as
`C_V=(3R)/2`
As gas is heated in a closed vessel i.e. at constant volume, if its temperature is raised from `T_1` to `T_2` then, we have heat supplied to the gas is
`Q=n C_V(T_2-T_1)`
or `2.49xx10^4=16/3xx3/2R(T_2-300)`
or `T_2=300+(3xx2xx2.49xx10^4)/(16xx3xx8.314)`
`=300+374.36`
=674.36 K
For constant volume process , we have
`P_2/P_1=T_2/T_1`
Thus `P_2=T_2/T_1xxP_1`
`=674.36/300xx1.6xx10^6`
`=3.6xx10^6 N//m^2`
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