A monoatomic gas undergoes a process given by `2dU+3dW=0`, then what is the process

Options

A. isobaric
B. adiabatic
C. isothermal
D. None of these

Answer

To determine the type of process a monoatomic gas undergoes given the relation \(2dU + 3dW = 0\), we can analyze the equation step by step.

### Step 1: Understand the given equation
The equation \(2dU + 3dW = 0\) can be rearranged to express the relationship between the change in internal energy (\(dU\)) and the work done (\(dW\)):
\[ 2dU = -3dW \]
This implies:
\[ dU = -\frac{3}{2} dW \]

### Step 2: Relate internal energy change to work done
In thermodynamics, for a monoatomic ideal gas, the change in internal energy (\(dU\)) is related to the temperature change (\(dT\)) by the equation:
\[ dU = \frac{3}{2} nR dT \]
where \(n\) is the number of moles and \(R\) is the universal gas constant.

### Step 3: Substitute \(dU\) in terms of \(dW\)
From the rearranged equation \(dU = -\frac{3}{2} dW\), we can substitute for \(dU\):
\[ -\frac{3}{2} dW = \frac{3}{2} nR dT \]
This leads to:
\[ dW = -nR dT \]

### Step 4: Analyze the implications of \(dW = -nR dT\)
The negative sign indicates that the work done by the gas is negative, which typically means that the gas is doing work on the surroundings (expansion). However, since the change in internal energy is also related to the temperature change, we can conclude that as the gas does work, its internal energy decreases, which suggests that the process is not isothermal (where \(dU = 0\)).

### Step 5: Identify the type of process
The relationship \(dU = -\frac{3}{2} dW\) indicates that the internal energy decreases as work is done. This behavior is characteristic of an **adiabatic process**, where there is no heat exchange with the surroundings, and the internal energy change is entirely due to work done.

### Conclusion
Thus, the process described by the equation \(2dU + 3dW = 0\) is an **adiabatic process**.

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