Calculate the value of mechanical equivalent of heat from the following data. Specific heat capacity of air at constant volume =`170 cal kg^(-1) K^(-1)` , `gamma = C_p/ C_v = 1.4` and the density of air at STP is `1.29 kg m^(-3)`. Gas constant `R = 8.3 JK^(-1) mol^(-1)`.

Explanation

Using `pV = nRT, ` the voluem of 1 mole of air at STP is
`V= (nRT)/(p) = ((1 mol) xx (8.3J //mol -K) xx (27 3 K))/(1.0xx10^(5) N//m ^(2))= 0.0224m ^(3)`
The mass of 1 mole is, therefore,
`(1.29 kg//m ^(3)) xx (0.0224m^(3))=0.029kg.`
The number of moles in 1 kg is ` (1)/(0.029).` The molar heat capacity at constant volume is
` C _(V) = (170cal)/((1//0.029)mol -K) =4.93cal//mol-K`
Hence `Cp = gamma C _(v) =1.4 xx 4.93 ` cal/mol-K
or, `C _(p) -C_(v) =0.4 xx 4.93 ` cal/mol-K `=1.97 cal/mol-K`
Thus, `8.3 J=1.97 cal.`
The mechanical equivalent of heat is
`(8.3J)/(1.97cal) = 4.2J//cal.`