Heat Capacity: Relation Between Cp and Cv

Introduction

Heat capacity or thermal capacity is described as the quantity of heat essential to alter the temperature of an item by 1 unit. Heat capacity is an intrinsic attribute of material & may alternatively be described as the ratio of the quantity of heat absorbed by a system to the shift in temperature.

Define Heat Capacity

When heat is absorbed by an object, the temperature rises; when heat is released, the temperature falls. The total kinetic energy of the components that comprise a body is measured by its temperature. When heat is absorbed by a body, it is transformed into kinetic energy of particle components, causing the temperature to rise. As a result, the temperature shift is proportional to the heat transmission.

The heat Q necessary to cause a variation in temperature ΔT of 1 mole of any substance is represented by the formula Q=mCΔT. The constant C here is referred to as the body's molar heat capacity. As a result, the molar heat capacity of any object is described as the quantity of heat energy needed to alter the temperature of 1 mole of that object by 1 unit. It is dependent on the system's type, size, & composition.

What are Heat Capacity C,Cp,and Cv?

Specific Heat (C)

Any material's specific heat capacity is described as "the amount of heat needed to alter the temperature of a unit mass of the material by 1 degree."

Q=mCΔT

As a result, specific heat capacity C=Q/mΔT

It is measured in joules per kelvin kilogram, or J/(KgK)

Constant Pressure (Cp)

The quantity of heat energy produced or absorbed by a unit mass of material with a shift in temperature at consistent pressure is defined as molar heat capacity at consistent pressure or Cp.

Under continual pressure, δQ=dU+PdV (isobaric process)

Cp can be expressed as −

$$\mathrm{Cp=[dH/dT]p}$$

where

The specific heat at consistent pressure is represented by Cp.

The enthalpy change is represented by dH.

The temperature shift is denoted by dT.

Constant Volume (Cv)

The molar heat capacity at a consistent volume abbreviated Cv is the quantity of heat released/absorbed per unit mass of an object at a consistent volume with a slight temperature variation.

With a consistent volume, dV=0,δQ=dU (isochoric process)

Cv can be written as −

$$\mathrm{Cv=[dU/dT]v}$$

Where,

Cv is the specific heat at a consistent volume, while dU is the tiny change in the system's internal energy.

Relationship Between Cp & Cv

The 1^st law of thermodynamics states that

Q=nCmΔT. . . . . . . . . . (1)

Equation 1 changes at consistent pressure P to

$$\mathrm{qP=nCpΔT}$$

This value equals the enthalpy change, i.e.

qP=nCpΔT=ΔH. . . . . . . . . . (2)

Similarly, with consistent volume V,

$$\mathrm{qV=nCvΔT}$$

This value equals the shift in internal energy, i.e.

qV=nCvΔT=ΔU. . . . . . . . . . . (3)

One mole (n=1) of an ideal gas,

$$\mathrm{ΔH=ΔU+Δ(PV)}$$

$$\mathrm{ΔH=ΔU+Δ(RT)}$$

Rearranging the previous equation yields

$$\mathrm{ΔH=ΔU+RΔT}$$

Therefore, ΔH=ΔU+RΔT

In the former equation, substitute the values of ΔH & ΔU from equations (2) & (3). (1),

$$\mathrm{nCpΔT=nCvΔT+RΔT}$$

The above equation may be expressed as follows for (n=1) −

$$\mathrm{CpΔT=CvΔT+RΔT}$$

Using ΔT as a common term, then

$$\mathrm{Cp×ΔT=(Cv+R)ΔT}$$

By removing the ΔT from both sides,

$$\mathrm{Cp=Cv+R}$$

$$\mathrm{Cp-Cv=R}$$

Why is Cp Greater than Cv?

The specific heats of an ideal gas are represented by the values Cp & Cv. These represent the quantity of heat essential to bring up the temperature of a unit mass by 1 degree.

According to the 1^st rule of thermodynamics,

$$\mathrm{ΔQ=ΔU+ΔW}$$

Where,

ΔQ is the quantity of heat introduced into the object, U is the shift in internal energy, & W is the quantity of work done.

At consistent pressure, heat is absorbed not just to raise internal energy, but also to accomplish work. At consistent volume, heat is absorbed just to raise internal energy & not to do any work on the system, as (for a closed system):

$$\mathrm{W=PΔV}$$

where W is the amount of work completed Here,

$$\mathrm{ΔV=0}$$

(A closed system is also 1 of the important criteria for consistent volume). As a result, the specific heat at consistent pressure exceeds the specific heat at a consistent volume, i.e.,

$$\mathrm{Cp>Cv }$$

Conclusion

The SI unit of molar heat capacity is the joule per kelvin mole (J/Kmol), often known as JK^-1 mol^-1. A calorimeter is a tool that measures heat. Since heat is delivered at consistent pressure, the molar heat capacity at consistent pressure Cp is always larger than the molar heat capacity at consistent volume Cv. Thermal expansion occurs when the size of an object grows with increasing temperature.

FAQs

1.What is the formula for calculating heat capacity utilising a supplied material's specific heat capacity?

Q=mCΔT

2.What does the International System of Units (SI) mean when it comes to specific heat capacity as well as heat capacity?

The SI units for heat capacity & specific heat capacity are joule per kelvin (J/K) & joule per kilogram Kelvin (J/KgK) respectively.

3.If we have 3.5 kg of water. What is the heat capacity of water if its specific heat is 4180 J/Kg⁰C?

$$\mathrm{c = 4180 \frac{J}{Kg}^\circ C}$$

m = 3.5 kg

$$\mathrm{Q=mC}$$

= 3.5×4180

$$\mathrm{= 14630 \frac{J}{^\circ C}}$$

4.A 600-gram cube of copper is heated at temperatures ranging from 30⁰C to 80⁰C. Compute the energy necessary to heat copper if its specific heat is stated as 0.385 J/g⁰C.

$$\mathrm{c = 0.385 \frac{J}{g}^\circ C}$$

$$\mathrm{ΔT = (80–30)^\circ C= 50^\circ C}$$

We are aware that, Q=mCΔT

= 600 × 0.385 × 50

= 11550 J

5.A metal ball weighing 35 grams is heated with 2000 J of energy at $\mathrm{100^\circ C}$. Determine the metal ball's specific heat.

m = 35 gm

$$\mathrm{ΔT = 100^\circ C}$$

Q = 2000 J

Including these values Q=mCΔT

$$\mathrm{2000 J = (35 g) c (100^\circ C)}$$

$$\mathrm{2000 J = (3500g^\circ C) c}$$

When both sides are divided by 3500 g⁰C

$$\mathrm{2000 J/ 3500^\circ C= c}$$

$$\mathrm{c = 0.571 \frac{J}{g}^\circ C}$$

6.What exactly is the distinction between specific heat capacity as well as heat capacity?

Specific heat capacity, as well as heat capacity, vary in a variety of ways. The primary distinction is that the specific heat capacity does not rely on the mass of the material, whereas heat capacity does.

7.The specific heat of 130 g iron is $\mathrm{0.45 J/g^\circ C}$. The iron item is heated at temperatures ranging from $\mathrm{100^\circ C\: to\: 500^\circ C}$. Determine the amount of heat energy necessary.

mass (m) = 130 g

specific heat of iron, (C) = $\mathrm{0.45\frac{J}{g}^\circ C}$

change in temperature (ΔT) = 500 – 100 = 400⁰C

$$\mathrm{Q=mCΔT}$$

$$\mathrm{Q = 130g × 0.45\frac{J}{g}^\circ C × 400^\circ C}$$

Q = 23400 J

As a result, 130 g of iron has a heat capacity of 23400 J.