
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let's determine the equation of the process. Since the piston is light, the elastic force of the spring at each moment in time is equal to the gas pressure force:

$$
k x = p(x) S
$$

where \(x\) is the length of the cylinder occupied by the gas, which is equal to the compression of the spring, \(k\) is the spring stiffness coefficient, and \(S\) is the cross-sectional area of the cylinder. By multiplying both parts of the equation by \(S\), we get

$$
p = \alpha V, \quad \alpha = \text{const.}
$$

Let's find the heat capacity of the gas in this process:

$$
c = \frac{dQ}{dT}
$$

Let the volume of gas \(V\) change by a small amount \(dV\). Then the pressure changes by \(dp = \alpha dV\), and the internal energy by

$$
dU = \frac{3}{2}((p + dp)(V + dV) - pV) = \frac{3}{2}(p dV + V dp) = 3 \alpha V dV
$$

In this case, the temperature change will be \(dT = \frac{2}{3} \frac{dU}{\nu R} = \frac{2 \alpha V dV}{\nu R}\).

The gas will perform work \(dA = p dV = \alpha V dV\).

The heat added to the system is equal to

$$
dQ = dU + dA = 4 \alpha V dV
$$

and the heat capacity is

$$
c = \frac{dQ}{dT} = 2 \nu R
$$

To find the specific heat capacity, it is necessary to divide the obtained heat capacity of the entire gas by its mass \(\nu \mu\), where \(\mu\) is the molar mass of the gas.

Present the answer in LaTex format: \boxed{Your answer}
user