Question

Ch 19 HW Relationships between Molar Heat Capacities 9 of 23 Constants The amount of heat needed to raise the temperature of 1 mole of a substance by one Celsius degree (or, equivalently, one kelvin) is called the molar heat capacity of the system, denoted by the letter C. If a small amount of heat dQ is put into n moles of a substance, and the resulting change in temperature for the system is dT, then Part A Consider an ideal gas being heated at constant volume, and let Cy be the gas's molar heat capacity at constant volume. If the gas's infinitesimal change in temperature is dT, find the infinitesimal change in internal energy dU of n moles of gas. Express the infinitesimal change in internal energy in terms of given quantities. This is the definition of molar heat capacity--the amount of heat Q added per infinitesimal change in T, per mole. View Available Hint(s) nν ΑΣφ A heated gas tends to expand, and the heat capacity depends on whether the gas is held at constant volume or allowed to expand so that it remains at constant pressure. The molar heat capacities are denoted C and C for constant volume and constant pressure, respectively When a gas is held at constant volume and heated, dV = 0, so the work, p dV done by the gas is zero. The first law of thermodynamics, dQ = dU + dW, therefore implies that all of the heat must go into increasing the gas's internal energy U. Submit When a gas is held at constant pressure as it is heated, the same amount of heat is required to increase the gas's internal energy. In addition, the gas expands and does work. The first law therefore demands addition of extra heat to do the work. Consequently, for a gas held at constant pressure, it takes more heat for a given increase in temperature than it does for a qas held at constant volume. Part B Complete previous part(s) Molar heat capacity at constant pressure This problem concerns the important relationship between molar heat capacity at constant volume C and the molar heat capacity at constant pressure Cp. Part C This discussion shows that the molar heat capacity of a qas depends on the circumstances under which the gas is heated. The molar heat capacity for a gas held at constant pressure is greater than that for a gas held at constant volume. In this problem, you will derive a single equation that relates Cp and Cy for an ideal gas. Now suppose the ideal gas is being heated while held at constant pressure p. The infinitesimal change in the gas's volume is dV, while its change in temperature is dT. Find the gas's molar heat capacity at constant pressure, Cp Express Cp in terms of some or all of the quantities Cy, p, dV, n, and dT. View Available Hint(s) ΠΠ ΑΣφ ? Сp 3 Submit

Answer 1

1 As given in question that it a small amount of heat do is put into n moles of a substance and the resulting change in temperature of the system is dT then Ic=1 do . C is molar heat Capacity. In dT I 1a) Ideal gas is heated at constant volume, the applying Ist Law of thermodynamics do = du + dw dg = du + Pdv s va constant} so, dg = du to given, ew the door onst, volume So, putting dg from equation (1) - Cv = 1 dl ha so, [du = new at this is change in internal energy.

Molar heat Capacity at constant pressure 2) Polar Iconst. P we know dQ = dut dw I this is the required expression for molar heat capacity at constant pressure. 1 3 Taking Breal ges we know that pV=MRT given p= const so, pdv tulp= nR aT Valle Vdp=0 TPav=nRat | Ist Law of thermodynamics do = dut dW dw =pdV=nRT . So, do= du tp dv=du taRaT |dg= dut RRIT we want to calculate Cp

q= 1 1 1 1 1 to, Ope the lead me ose the le tt ne] we know that (dua heq do] So, co = to new to the [p= eta ] we know that putting & . GTR we know that pdv =nRdT So, R = par ndt this is the required expression. Already disscussed above 4) 16 given that opar Op=6tR But we know Cyth Y= l+R