STUDY4MECHANICAL for Mechanical Engineers

In this article, you will learn the relations between two specific heats Cp and Cv.  Relations between specific heats:  The difference between two specific heats is equal to a gas constant.  Cp - Cv = R  The ratio of two specific heats is known as an adiabatic index and it denoted as ϒ. We know above equation, Cp - Cv = R  Cp / Cv = 1 + R / Cv  Now we take Cp / Cv =  ϒ ϒ = 1 + R / Cv As we have seen in the article  Why is the specific heat at constant pressure greater than specific heat at constant volume? We all know that Cp is always greater than Cv therefore the value of  ϒ is always greater than unity.  The value of  ϒ for air is 1.4. 

Basic Explanation:  When a gas is heated at constant pressure, the heat supplied to the gas is utilized in the following ways.  To raise the temperature of the gas. The heat remains within the body and represents the increase in internal energy. dU = m Cv (T2-T1)  To do some external work during expansion. W = p (V2-V1) = m R (T2-T1) Thus, specific heat at constant pressure is greater the specific heat at constant volume. Mathematical Explanation:  According to the first law of thermodynamics   Q = ΔU + W Where, (W= P ΔV ) Internal energy measures the temperature of the body.  At constant volume as there is no work done on the gas so  ΔV = 0. All the heat supplied utilize an increase in internal energy. Hence, the temperature of the body increases with less supply of heat. Whereas, at constant pressure, the supply of heat accounts for both internal energy and gas work. Thus, more heat is required to increase the temperature.  Thus, Cp >

Definition of specific heat: The amount of heat per unit mass required to raise the temperature by one degree Celcius is called specific heat. It is denoted as c. All the solids and liquids have only one specific heat and gas can have any number of specific heat depending upon conditions, under which it is heated.  Types of specific heat: Specific heat at constant volume  Specific heat at constant pressure Specific heat at constant volume: The amount of heat required to raise the temperature of a unit mass of gas by one degree Celcius is called specific heat at constant volume. It is denoted by Cv.  Formula: The total heat supplied to the gas at constant volume is given below.  Q = Mass × Rise in temperature × Specific heat at constant volume Q = m ×(T2 - T1) × Cv  Where m = mass of the gas T1 = Initial temperature of the gas T2 = Final temperature of the gas When specific heat at constant volume Cv is multiplied by the molecular

The universal gas constant is also called molar constant. It is generally denoted as Ru. It is the product of a gas constant and the molecular mass of the gas.  Ru = R M  Where R = Gas constant M = Molecular mass of gas expressed in kg mol In general, we take the different molecular mass of the different gas such as M1, M2, M3 and R1, R2, R3 are their gas constant respectively. So, we conclude that  M1 R1 = M2 R2 = M3 R3 Important Notes:  The value of Ru is same for all the gas.  SI Unit of Ru is 8314 J / kg mol K or 8.314 kJ / kg mol K 

The principle of Joule's law is that at a constant temperature the internal energy of an ideal gas is independent of volume.  Statement of Joule's Law: The change of internal energy of a perfect gas is directly proportional to the change of temperature.  Let we see above statement in mathematical form. Formula: dE ∞ dT or dE = m c dT Where m = mass of the gas c = Constant of proportionality, also known as specific heat Notes: We see that its internal energy, as well as temperature, does not change as the gas expands, without doing any external work and without taking in or giving our heat.

Some important point on the difference between characteristics gas constant and universal gas constant are explained below. Let us check out one by one below. You can also check out characteristics equation of gas , and general gas equation to know more about it. Difference between characteristics gas constant and universal gas constant: Constant gas characteristics vary from gas to gas but the constant of universal gas is fixed. If you working at human scales, then use universal gas constant, whereas if you want to gas constant that defined per unit mass of a specific gas, then use the specific gas constant.  The characteristic gas constant in equation P = RT is the value of R for a given gas for use. It is equal to the constant of universal gas divided by the molar mass of the particular gas. Characteristics gas Constant applicable for an ideal gas, whereas universal gas constant applicable for a real gas.  Units: Characteristics gas constant - k J / kg K and Univers

Characteristics equation of a gas is the modified equation of general gas. If the volume in a general gas is taken as 1 kg of gas, then the constant C is represented by another constant R.  Thus, the general gas equation may be rewritten as follows.  P V = R T R = Characteristics gas constant or Simply gas constant For any mass m kg of gas, the equation may be rewritten as follows. m P V = R T OR P V = m R T P = m / V  ×  R T Let we take m / V = ƿ  P = ƿ R T Where, P = Pressure ( 1 bar = 100  KN/m 2 ) R = Gas constant (287 J / kg K = 0287 k J / kg K) ƿ = Density of the given gas