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**Hint:**The answer here to the question is based on the concept of physical chemistry that includes that molar specific heat capacity ${{C}_{v}}$ is given for constant volume and is applied for ideal gas. The formula contains an answer.

**Complete step by step solution:**

In our basic concepts of chemistry, we have come across the topics in physical chemistry about the specific heat capacity, latent heat of vaporisation and various terms relating to this.

Let us now see what is meant by specific heat capacity and its measurement.

- Specific heat capacity is defined as the total amount of heat required to raise the temperature of one kg of a substance by one Kelvin.

- The specific heat capacity has the unit J/kg/K and this is measured as it provides the means to relate the amount of thermal energy gained or lost by a sample of any substance to that of the sample's mass and the resulting change in the temperature mathematically.

Therefore, based on these facts we can say that the ideal gases have the specific heat capacity in combination with the real gas constant ‘R’ based on type of gas that is,

- For a monatomic gas it is given by $Cv=\dfrac{f}{2}R$ where f is the degree of freedom and for monoatomic gas, f = 3.

Thus, it has ${{C}_{v}}=\dfrac{3}{2}R$

Similarly, diatomic gas has degrees of freedom f = 5. Therefore, the specific heat capacity for diatomic gas is${{C}_{v}}=\dfrac{5}{2}R$

Therefore, the correct answer is ${{C}_{v}}$ values for monoatomic and diatomic gases are $\dfrac{3}{2}R$ and $\dfrac{5}{2}R$ respectively.

**Note:**Note that for solids and most liquids the value of specific heat capacity at constant pressure ${{C}_{p}}\sim {{C}_{v}}$ and this fact also helps to solve the questions relating to specific heat capacity and do not confuse.

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