A vessel is filled with a gas at a pressure of 76 cm of Hg at some temperature. The mass of the gas is increased by 50% by introducing more gas in the vessel at the same temperature. Calculate the new pressure of the gas.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the initial conditions
We know that the initial pressure \( P_1 \) of the gas is 76 cm of Hg. Let's denote the initial mass of the gas as \( m_1 \).

### Step 2: Calculate the new mass of the gas
The mass of the gas is increased by 50%. Therefore, the new mass \( m_2 \) can be calculated as:
\[ m_2 = m_1 + 0.5 m_1 = 1.5 m_1 \]

### Step 3: Use the ideal gas law
According to the ideal gas law, the relationship between pressure, volume, and mass (at constant temperature) can be expressed as:
\[ PV = mRT \]
Where:
- \( P \) is the pressure
- \( V \) is the volume (constant in this case)
- \( m \) is the mass of the gas
- \( R \) is the gas constant
- \( T \) is the temperature (constant in this case)

### Step 4: Establish the relationship between pressures and masses
Since the volume and temperature are constant, we can say that the pressure is directly proportional to the mass of the gas:
\[ \frac{P_2}{P_1} = \frac{m_2}{m_1} \]

### Step 5: Substitute the values
Substituting \( m_2 = 1.5 m_1 \) into the equation gives:
\[ \frac{P_2}{P_1} = \frac{1.5 m_1}{m_1} \]
This simplifies to:
\[ \frac{P_2}{P_1} = 1.5 \]

### Step 6: Calculate the new pressure
Now, we can find \( P_2 \):
\[ P_2 = 1.5 \times P_1 = 1.5 \times 76 \, \text{cm of Hg} \]
Calculating this gives:
\[ P_2 = 114 \, \text{cm of Hg} \]

### Final Answer
The new pressure of the gas is \( P_2 = 114 \, \text{cm of Hg} \).

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