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(a) Define two specific heats of a gas. Why is `C_(p) gt C_(v)`? (b) Shown that for an ideal gas, `C_(p) = C_(v) +(R )/(J)` |
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Answer» Correct Answer - (a) The specific heat capacity of a substance is defined as the heat supplied per unit mass of the substance per unit rise in the temperature. 1. Specific heat of gas at constant volume. The specific heat capacity of a gas is defined as the heat given per gram of the gas per unit rise in the temperature at constant volume. It is denoted by `C_(v)`. MOlar specific heat a constant volume. The molar heat capacity of a gas is defined as the heat given per mole of teh gas per unit rise in the temperature at constant volume. it is denoted by `C_(v)`. `C_(v) = M C_(v)` Here `M` is the molecular weight of the gas. 2. Specific heat of a gas at constant pressure. It is defined as the amount of heat required to raise the temperature of `1g` of a gas through `1^(@)C` at constant pressure. It is denoted by `C_(p)`. Molar specific heat of a gas at constant pressure. Molar specific heat of a gas at constant pressure. It is defined as the amount of heat required to raise the temperature of `1` mole of a gas through `1^(@)C` at constant pressure. It is denoted by `C_(p)`. Obviously, `C_(p) = M C_(p)` Here `M` is the molecular weight of the gas `C_(p)` is greater than `C_(v)` When a gas is heated both its volume and pressure change. Let us estimate the amount of heat required by the gas to heat it thorugh `1^(@)C`, when either its volume or its pressure is kept constant. (i) When volume is kept constant. The pressure on the gas is increased so that its volume remains constant. As the gas cannot perform work `(V =` constant), the heat supplied will increase only the temperature or the internal energy of the gas. Therefore, in case of `C_(v)`, heat is required only for increasing the temperature of the gas through `1^(@)C`. (ii) When pressure is kept constant. When the gas is heated at constant pressure, if expands alos. Therefore, the heat sullpied at constant pressure will partly increase its tempertaure (internal energy) and partly will be utillised in performing work aganist teh external pressure. Therefore in case of `C_(p)` more heat (as compared to the case of `C_(v))` will be required for increasing the temperature of the gas through `1^(@)C`. Hence the specific heat of a gas at constant pressure is greater then the specific heat at constant volume i.e., `C_(p) kt C_(v)`. The difference between the values of two specific heats is equal to the amount fo heat equivalent to the work performed by the gas during expansion at constant pressure. (b) The relation between two specific heats of a gas can be derived by applying first law of thermodynamics. Consider one mole of an ideal gas. Suppose that the gas is heated at constant volume, so that its temperature increases by `dT`. If `dQ` is the amount of heat supplied, then `dQ = 1 xx C_(v) xx dT` or `dQ = C_(v) dT ..........(i)` As the gas is heated at constant volume it will not performs any external work and in accordance with the first law of thermodynamics. `dQ = dU +0 = dU` Setting `dQ = dU` in the equation (i) we have `dU = C_(v)dT ..........(ii)` Now, heat the gas at constant pressure so as to again increase it temperature by `dT`. If `dQ` is the amount of heat supplied then `dQ = 1 xx C_(v) xx dT` or `dQ = C_(v) xx dT .........(iii)` the hea supplied at constant pressure increases the temperature of teh gas by `dT` i.e., increases its internal energy by `dU` and as well as enables the gas to perform work `dW`. if `dV` is the increase in volume of the gas, the work performed by the gas, `dW = P dV .........(iv)` Accoding to teh first law of thermodynamics, the total heat supplied to the gas to heat it at constant pressure, `dQ = dU +dW` Using the equations (i), (ii) and (iii) we get `C_(p) dT = C_(v) dT +P dV ......(v)` For one mole of a perfect gas, `PV = RT` The heat `dQ` is supplied to the gas at constant pressure `P`. Therefore, defferentiating both sides of the above equation w.r.t. by treating `P` as constant, `(d)/(dT) (PV) =(d)/(dT) (RT) or P(dV)/(dT) = R` or `PdV = R dT ..........(vi)` From the equations (v) and (vi) we have `C_(p)dT = C_(u)dT +R dT` or `C_(p) = C_(v) +R` |
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