Assume the gas to be ideal and monotomic.
In theory the specific heat capacity of a mole of gas can have any value between $-\infty$ and $+\infty$ depending on the heat transfer in/out of the gas and the work done by/on the gas.
The defining equation for specific heat capacity is $Q=n\,C\,\Delta T$ where $Q$ is the heat transferred, $n$ is the number of moles of gas, $C$ is specific heat capacity of the gas and $\Delta T$ is the change in temperature of the gas.
The first law of thermodynamics $\Delta U = Q -W$ where $\Delta U$ is the change in internal energy of the gas, $Q$ is the heat transferred into the gas and $W$ is the work done by the gas shows that you can change the internal energy of the gas, which is dependent on the temperature of the gas, by an infinite number of permutations of $Q$ and $W$ thus allowing there to be an infinite number of specific heat capacities of a gas.
In practice for convenience the specific heat capacity is defined; at constant volume ie $W=0$ and then $\Delta U = Q$ and at constant pressure ie $W=P\Delta V$ where $\Delta V$ is the change in volume of the gas and then $\Delta U = Q-P\Delta V$.
As an example one could compress a gas in a chamber which allows no heat in or out of the chamber.
Work is done on the gas and so its temperature rises with no transfer of heat and the specific heat capacity of the gas is zero, $0=n\,C\, \Delta T$.
Another example might be of a gas which is being heated at a certain rate and the gas is doing work at the same rate ie $Q=W$.
There is no change in the internal energy of the gas $\Delta U = Q-W =0$ and so the gas does not change its temperature so the gas has an infinite specific heat capacity, $Q = n\,C\, 0$.
why do Cv values of same substance at different volume are equal since the formula of Cv has nothing to do with volume
Because if all else is equal (temperature and pressure) the number of moles of gas depends on the volume of the gas so if you have twice the volume of gas you have twice the number of moles but the value of the specific heat capacity of the gas has not changed even though twice the amount of heat is required to change the temperature of the gas by the same amount.
I think you mean to say that they have a different number of particles- 1kg of lead and 1kg of copper surely have the same mass.
There are $~2.91\times10^{24}$ atoms in a kg of lead, and $~9.45\times10^{24}$ atoms in a kg of copper. Heat (more accurately, thermal energy) is "stored" in the particular degrees of freedom for the motion of the atomic "particles"- and since there are more atoms in a kilogram of copper, it's gonna have more atoms with more degrees of freedom. Thus, more capacity for storage.
Best Answer
Primarily because 1 kg of water has more atoms than 1 kg of copper.
For ordinary solids at or above room temperature, the molar heat capacity is approximately the same (Dulong and Petit's law), three times the gas constant, about 25 J/K. It is because the sum of potential and kinetic energy per atom is $3k_BT$ in the harmonic approximation.
Hydrogen in water or ice is a bit different. It is so light that quantum effects come inte play, equipartition does not apply.