Temperature of a monoatomic gas is increased from `T_0` to `2T_0` in three different processes:
isochoric, isobaric and adiabatic. Heat given to the gas in these three processes are `Q_1, Q_2 and Q_3` respectively. Then, choose the correct option.

Options

A. (a) `Q_1gtQ_3`
B. (b) `Q_2gtQ_1`
C. (c) `Q_2gtQ_3`
D. (d) `Q_3=0`

Explanation

To solve the problem, we need to analyze the heat added to a monoatomic gas during three different processes: isochoric, isobaric, and adiabatic. We will denote the heat added in these processes as \( Q_1 \), \( Q_2 \), and \( Q_3 \) respectively.

### Step 1: Heat in Isochoric Process
In an isochoric process (constant volume), the heat added to the gas can be calculated using the formula:
\[ Q_1 = n C_v \Delta T \]
where:
- \( n \) = number of moles of the gas
- \( C_v \) = molar specific heat at constant volume
- \( \Delta T \) = change in temperature

Given that the temperature increases from \( T_0 \) to \( 2T_0 \), we have:
\[ \Delta T = 2T_0 - T_0 = T_0 \]
Thus,
\[ Q_1 = n C_v T_0 \]

### Step 2: Heat in Isobaric Process
In an isobaric process (constant pressure), the heat added can be calculated using the formula:
\[ Q_2 = n C_p \Delta T \]
where \( C_p \) is the molar specific heat at constant pressure. Using the same change in temperature:
\[ Q_2 = n C_p T_0 \]

### Step 3: Heat in Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. Therefore, the heat added to the gas is:
\[ Q_3 = 0 \]

### Step 4: Comparing the Heats
Now, we can compare the heats \( Q_1 \), \( Q_2 \), and \( Q_3 \):
1. Since \( C_p > C_v \), it follows that:
\[ Q_2 > Q_1 \quad \text{(since \( n C_p T_0 > n C_v T_0 \))} \]
2. Since \( Q_3 = 0 \), we have:
\[ Q_1 > Q_3 \quad \text{(since \( Q_1 = n C_v T_0 > 0 \))} \]

### Conclusion
From the above comparisons, we can summarize the relationships as follows:
\[ Q_2 > Q_1 > Q_3 \]
or in terms of the values:
\[ Q_2 > Q_1 > 0 \]

### Final Answer
Thus, the correct relations among the heats are:
- \( Q_2 > Q_1 \)
- \( Q_1 > Q_3 \) (where \( Q_3 = 0 \))