Chemistry Applications-II

### Topics Covered :

● Enthalpy, H
● Extensive and Intensive Properties
● Heat Capacity
● The Relationship between C_p and C_V of an Ideal Gas

### Enthalpy, H (A useful new state function) :

=> The heat absorbed at constant volume is equal to change in the internal energy i.e., color{purple}(ΔU = q_V).

● But most of chemical reactions are carried out not at constant volume, but in flasks or test tubes under constant atmospheric pressure. So, we need to define another state function which may be suitable under these conditions.

=> We may write equation (6.1) as color{purple}(ΔU = q_p− pΔV) at constant pressure, where q_p is heat absorbed by the system and color{purple}(–pΔV) represent expansion work done by the system.

● Let us represent the initial state by subscript 1 and final state by 2. We can rewrite the above equation as

color{purple}(U_2–U_1 = q_p – p (V_2 – V_1)

● On rearranging, we get

color{purple}(q_p = (U_2 + pV_2) – (U_1 + pV_1)) ........(6.6)

Now we can define another thermodynamic function, the enthalpy H [Greek word enthalpien, to warm or heat content] as :

color{purple}(H = U + pV) .................... (6.7)

So, equation (6.6) becomes

color{purple}(q_p= H_2 – H_1 = ΔH)

=> Although color{purple}(q) is a path dependent function, color{purple}(H) is a state function because it depends on U, p and V, all of which are state functions.

● Therefore, color{purple}(ΔH) is independent of path. Hence, color{purple}(q_p) is also independent of path.

=> For finite changes at constant pressure, we can write equation 6.7 as

color{purple}(ΔH = ΔU + ΔpV)

Since color{purple}(p) is constant, we can write

color{purple}(ΔH = ΔU + pΔV) .................... (6.8)

color{red}("Note ") When heat is absorbed by the system at constant pressure, we are actually measuring changes in the enthalpy.

=> color{purple}(ΔH = q_p), heat absorbed by the system at constant pressure.

● color{purple}(ΔH) is negative for color{red}("exothermic reactions") which evolve heat during the reaction

● color{purple}(ΔH) is positive for color{red}("endothermic reactions") which absorb heat from the surroundings.

=> At constant volume color{purple}((ΔV = 0), ΔU = q_V), therefore equation 6.8 becomes

color{purple}(ΔH = ΔU = q_V)

=> The difference between color{purple}(ΔH) and color{purple}(ΔU) is not usually significant for systems consisting of only solids and/or liquids. Solids and liquids do not suffer any significant volume changes upon heating.

=> The difference becomes significant when gases are involved.

● Let us consider a reaction involving gases.

● If color{purple}(V_A) is the total volume of the gaseous reactants, color{purple}(V_B) is the total volume of the gaseous products, n_A is the number of moles of gaseous reactants and color{purple}(n_B) is the number of moles of gaseous products, all at constant pressure and temperature, then using the ideal gas law, we write,

color{purple}(pV_A = n_ART) and color{purple}(pV_B = n_BRT)

Thus, color{purple}(pV_B – pV_A = n_BRT – n_ART = (n_B–n_A)RT)

or color{purple}(p (V_B – V_A) = (n_B – n_A) RT)

or color{purple}(p ΔV = Δn_gRT) ................. (6.9)

● Here, color{purple}(Δn_g) refers to the number of moles of gaseous products minus the number of moles of gaseous reactants.

● Substituting the value of color{purple}(pΔV) from equation 6.9 in equation 6.8, we get

color{purple}(ΔH = ΔU + Δn_gRT) ................ (6.10)

The equation 6.10 is useful for calculating color{purple}(ΔH) from color{purple}(ΔU) and vice versa.
Q 3047623583

If water vapour is assumed to be a perfect gas, molar enthalpy change for vapourisation of 1 mol of water at 1bar and 100°C is 41kJ mol^(–1). Calculate the internal energy change, when

(i) 1 mol of water is vaporised at 1 bar pressure and 100°C.
(ii) 1 mol of water is converted into ice.

Solution:

(i) The change H_2O (l) → H_2O (g)

DeltaH = DeltaU+ Deltan_g RT

or DeltaU = DeltaH – Deltan_gR T , substituting the values, we get

DeltaU = 41.00kJ mol^(-1) -1 xx8.3 J mol^(-1) K^(-1) xx 373K

 = 41.00kJ mol^(-1) -3.096 kJ mol^(-1)

 = 37.904 kJ mol^(-1)

(ii) The change H_2O (l) → H_2O (s)

There is negligible change in volume, So, we can put p Delta V = Deltan_g RT approx O in this case DeltaH approx DeltaU

so DeltaU = 41.00kJ mol^(-1)

### Extensive and Intensive Properties :

=> In thermodynamics, a distinction is made between extensive properties and intensive properties.

color{green}("Extensive Property ") An extensive property is a property whose value depends on the quantity or size of matter present in the system.

color{red}("Example ") Mass, volume, internal energy, enthalpy, heat capacity etc. are extensive properties.

color{green}("Intensive Properties ") Those properties which do not depend on the quantity or size of matter present are known as intensive properties.

color{red}("Example ") Temperature, density, pressure etc. are intensive properties.

=> A molar property, color{purple}(χ_m), is the value of an extensive property color{purple}(χ) of the system for 1 mol of the substance.

● If n is the amount of matter, color{purple}(χ_m = χ/n) is independent of the amount of matter.

● Other examples are molar volume, color{purple}(V_m) and molar heat capacity, color{purple}(C_m).

=> The distinction between extensive and intensive properties can be understood by considering a gas enclosed in a container of volume color{purple}(V) and at temperature color{purple}(T) [Fig. 6.6(a)].

● Let us make a partition such that volume is halved, each part [Fig. 6.6 (b)] now has one half of the original volume, V/2, but the temperature will still remain the same i.e., color{purple}(T).

● It is clear that volume is an extensive property and temperature is an intensive property.

### Heat Capacity :

=> In this section, let us see how to measure heat transferred to a system. This heat appears as a rise in temperature of the system in case of heat absorbed by the system.

● The increase of temperature is proportional to the heat transferred color{purple}(q = text(coeff) × ΔT)

● The magnitude of the coefficient depends on the size, composition and nature of the system.

● We can also write it as color{purple}(q = C ΔT)

● The coefficient, color{purple}(C) is called the color{red}("heat capacity").

● Thus, we can measure the heat supplied by monitoring the temperature rise, provided we know the heat capacity.

● When color{purple}(C) is large, a given amount of heat results in only a small temperature rise. Water has a large heat capacity i.e., a lot of energy is needed to raise its temperature.

color{purple}(C) is directly proportional to amount of substance. The molar heat capacity of a substance, C_m = C/n is the heat capacity for one mole of the substance and is the quantity of heat needed to raise the temperature of one mole by one degree celsius (or one kelvin).

color{green}("Specific Heat Capacity" ) Specific heat, also called specific heat capacity is the quantity of heat required to raise the temperature of one unit mass of a substance by one degree celsius (or one kelvin).

● For finding out the heat, color{purple}(q), required to raise the temperatures of a sample, we multiply the specific heat of the substance, color{purple}(c), by the mass color{purple}(m), and temperatures change, color{purple}(ΔT) as

color{purple}(q = c xx m xx Delta T) .............(6.11)

### The relationship between color{purple}(C_p) and color{purple}(C_V) for an ideal gas :

=> At constant volume, the heat capacity, color{purple}(C) is denoted by color{purple}(C_V) and at constant pressure, this is denoted by color{purple}(C_p).

● Let us find the relationship between the two.

=> We can write equation for heat, color{purple}(q)

● at constant volume as color{purple}(q_V = C_VΔT = ΔU)

● at constant pressure as color{purple}(q_p = C_p DeltaT = Delta H)

=> The difference between color{purple}(C_p) and color{purple}(C_V) can be derived for an ideal gas as :

● For a mole of an ideal gas, color{purple}(ΔH = ΔU + Δ(pV ))

color{purple}(= ΔU + Δ(RT ))

color{purple}(= ΔU + RΔT)

∴ color{purple}(ΔH = ΔU + RΔT) ................... (6.12)

On putting the values of color{purple}(ΔH) and color{purple}(ΔU), we have

color{purple}(C_pΔT = C_VΔT+RΔT)

color{purple}(C_p = C_v + R)

color{purple}(C_p -C_v = R) ......(6.13)