# Enthalpy

**Enthalpy (H)**

- Heat absorbed at constant volume is equal to the change in internal energy.
- Most chemical reactions occur under constant atmospheric pressure, so we need to define another state function.
- We can represent the initial and final states using subscript 1 and 2, respectively.
- Enthalpy (H) is defined as U + pV, where U is internal energy, p is pressure, and V is volume.
- Enthalpy is a state function and is independent of path, while q is a path dependent function.
- At constant pressure, qp (heat absorbed by the system) is equal to ∆H (change in enthalpy).
- For finite changes at constant pressure, ∆H = ∆U + p∆V.
- When solids and liquids are involved, the difference between ∆H and ∆U is not significant, but it becomes significant when gases are involved.
- For a reaction involving gases, we can use the ideal gas law to calculate p∆V and ∆ng (change in the number of moles of gaseous products minus the number of moles of gaseous reactants).
- ∆H = ∆U + ∆ngRT, where R is the gas constant and T is temperature.

**Extensive property**

An extensive property is a property that depends on the quantity or size of matter present in the system. Examples of extensive properties include mass, volume, internal energy, enthalpy, heat capacity, etc. The value of an extensive property increases with the amount of matter in the system. For instance, if the amount of matter in a system is doubled, then the value of the extensive property will also double.

**Intensive property**

An intensive property is a property that does not depend on the quantity or size of matter present in the system. Examples of intensive properties include temperature, pressure, density, refractive index, etc. The value of an intensive property remains the same regardless of the amount of matter in the system. For instance, if the amount of matter in a system is doubled, the value of the intensive property will remain the same.

**Heat capacity**

Heat capacity is the amount of heat required to raise the temperature of a substance by one degree Celsius (or one Kelvin). It is represented by the symbol C and has units of J/°C (joules per degree Celsius) or J/K (joules per Kelvin).

The formula for heat capacity is:

C = q / ∆T

Where:

- C = heat capacity
- q = amount of heat transferred to the system
- ∆T = change in temperature of the system

In other words, the heat capacity is the ratio of the heat transferred to the change in temperature. It is a measure of how much heat a substance can absorb before its temperature starts to increase. The larger the heat capacity, the more heat energy is required to raise the temperature of the substance.

Molar Heat Capacity: The molar heat capacity of a substance, denoted by Cm, is the amount of heat energy required to raise the temperature of one mole of the substance by one degree Celsius (or Kelvin). It is expressed in units of J/(mol.K) and is given by the formula:

C_{m} = q/(n∆T)

where q is the amount of heat energy absorbed, n is the number of moles of the substance, and ∆T is the change in temperature.

Specific Heat Capacity: The specific heat capacity of a substance, denoted by c, is the amount of heat energy required to raise the temperature of one unit mass of the substance by one degree Celsius (or Kelvin). It is expressed in units of J/(g.K) and is given by the formula:

c = q/(m∆T)

where q is the amount of heat energy absorbed, m is the mass of the substance, and ∆T is the change in temperature.

The specific heat capacity is related to the molar heat capacity by the equation:

C_{m} = c × M

where M is the molar mass of the substance.

**Relationship between Cp and Cv for ideal gas**

The relationship between the specific heat capacities at constant pressure (C_{P}) and constant volume (Cv) for an ideal gas can be derived using the first law of thermodynamics and the definition of these two specific heat capacities.

The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system:

ΔU = Q - W

For an ideal gas, the internal energy is a function of temperature only, and therefore:

dU = Cv dT

where Cv is the molar heat capacity at constant volume.

The work done by the system can be expressed in terms of pressure and volume:

W = -PΔV

where P is the pressure and ΔV is the change in volume.

The heat added to the system at constant volume is given by:

Qv = Cv ΔT

where ΔT is the change in temperature.

At constant pressure, the heat added to the system is given by:

Qp = C_{P} ΔT

where CP is the molar heat capacity at constant pressure.

Therefore, using the first law of thermodynamics:

ΔU = Qp - W

Substituting the expressions for Qp and W:

Cp ΔT - PΔV = Cv ΔT

Since the gas is assumed to be ideal, we can use the ideal gas law:

PV = nRT

where n is the number of moles, R is the gas constant, and T is the temperature.

Differentiating both sides of the ideal gas law with respect to volume at constant temperature:

PdV + VdP = nRdT

Dividing by V and rearranging:

dP/dT = nR/V

Substituting into the expression for ΔV:

ΔV = nRΔT/P

Substituting into the expression for ΔU:

Cp ΔT - nRΔT = Cv ΔT

Simplifying:

Cp - Cv = nR

Therefore, the relationship between the specific heat capacities at constant pressure and constant volume for an ideal gas is:

C_{P} - C_{V} = nR

This relationship holds for all ideal gases, regardless of their specific heat capacity values.