establish relation between two principle specific heat of a gas
Answers
Specific heat capacity is the amount of heat required to raise the temperature of substance with unit mass through unit temperature. i.e. Q = m. s. dT
when, m = dT = 1, Q = s
In case of gas, the heat required is different for the same raise in temperature under:
i) constant volume
ii) constant pressure
It is because temperature of a gas is the average kinetic energy of the its molecules. Raising in temperature of body means raising the average kinetic energy of body (which in turn means more jiggling of its atoms or molecules).
Raise in temperature i.e average kinetic energy, under constant volume increases the pressure of the gas. However, raise in temperature under constant pressure, the volume of the gas must increase (in-order to maintain constant pressure). So, more energy is required in increasing the volume of the gas (under constant pressure).
Thus, Sp>Sv
where, Sp = the specific heat capacity of a gas under constant pressure
Sv = the specific heat capacity of gas under constant volume
Answer:
this is your answer thankyou for the question#be happy
Explanation:
To study the relation with degrees of freedom: The heat capacity ratio (gamma, γ) for an ideal gas can be related to the degrees of freedom ( f ) of gas molecules by the formula: γ = 1 + 2f2f\frac{2}{f}, or f = 2γ−12γ−1\frac{2}{γ-1}The specific heat of gas at constant volume in terms of degree of freedom 'f' is given as: Cv = (f/2) R.Also, Cp - Cv = R Therefore, Cp = (f/2) R + R =R (1 + f/2)Now, ratio of specific heats γ is given as:γ = Cp/Cv = R(1+f/2)/(f/2)R(f/2)R(f/2) R =(2+f)/f Or, γ = 1+ 2/f So, we can also say that, Cp/Cv = (1 + 2/f), where f is degree of freedom.Monoatomic gas has only one translational motion, hence three translational degrees of freedom.The average energy of the molecule E, at temperature T, is given by: E= ½ mvx2 + 1/2mvy2 + 1/2mvz2where, m is the mass of the molecule, v (vx, vy and vz) is the momentum of a molecule along x-axis, y-axis and z-axis. The Law of Equipartition Gives: E= ½ kBT + ½ kBT + ½ kBT= 3/2kBT; here, kB is average translational kinetic energy, T is the temperature. Energy per molecule of gas is given by: U=3/2kBT = 3/2 RTThus, dU/dT = 3/2 RBut, dU/dT= CvThus, Cv = 3/2 RNow, Cp - C v = RThen, Cp = R + Cv That gives, Cp = R + 3/2 R = 5/2 Rγ = Cp /Cv = (5/2 R )/ (3/2R) = 5/3 = 1.67For a diatomic gas (such as, H₂, O₂ and N₂), has 5 as degrees of freedom (3 as translational and 2 as rotational degrees of freedom at room temperature; whereas, except at high temperatures, the vibrational degree of freedom is not involved). Why is Cp Greater than Cv?The values indicated by Cp and Cv are the specific heats of an ideal gas. These indicate the quantity of heat that can increase the temperature of unit mass by 1°C.By the first law of thermodynamics, ΔQ = ΔU + ΔWwhere, ΔQ is the amount of heat that is given to the system, ΔU is the change in internal energy and ΔW is the work done.So, at constant pressure, heat is absorbed not only for increasing the internal energy (function of temperature) but for doing work as well. Whereas, at constant volume, heat is absorbed only for raising the internal energy and not for doing any kind of work on the system as (for a closed system): W = PΔV, where W is the work done. Here, ∆V = 0. (A closed system is also one of the crucial conditions for constant volume). Hence, the specific heat at a constant pressure is more than specific heat at a constant volume, i.e. Cp > Cv.