The density of liquid `CO_(2)` at room temperautre is `"0.8 g cm"^(-3)`. How large a cartridge of liquid `CO_(2)` must be provided to inflate a life jacket of 4 litres capacity at S.T.P.?

Explanation

To solve the problem of determining how large a cartridge of liquid CO₂ must be provided to inflate a life jacket of 4 liters capacity at standard temperature and pressure (STP), we can follow these steps:

### Step 1: Understand the given information
- The density of liquid CO₂ at room temperature is \(0.8 \, \text{g/cm}^3\).
- The volume of the life jacket is \(4 \, \text{liters}\).
- At STP (Standard Temperature and Pressure), the conditions are:
- Temperature: \(0^\circ C\) or \(273 \, \text{K}\)
- Pressure: \(1 \, \text{bar}\) or \(1 \, \text{atm}\)

### Step 2: Convert the volume of the life jacket to cubic centimeters
Since the density is given in g/cm³, we need to convert the volume from liters to cm³:
\[ 4 \, \text{liters} = 4 \times 1000 \, \text{cm}^3 = 4000 \, \text{cm}^3 \]

### Step 3: Use the Ideal Gas Law to find the mass of CO₂ needed
The Ideal Gas Law is given by:
\[ PV = nRT \]
Where:
- \(P\) = pressure (1 atm)
- \(V\) = volume (4000 cm³)
- \(n\) = number of moles
- \(R\) = ideal gas constant (0.0821 L·atm/(K·mol))
- \(T\) = temperature (273 K)

First, convert the volume from cm³ to liters:
\[ 4000 \, \text{cm}^3 = 4 \, \text{liters} \]

Now, substituting the values into the Ideal Gas Law:
\[ (1 \, \text{atm}) \times (4 \, \text{L}) = n \times (0.0821 \, \text{L·atm/(K·mol)}) \times (273 \, \text{K}) \]

Solving for \(n\):
\[ n = \frac{(1 \, \text{atm}) \times (4 \, \text{L})}{(0.0821 \, \text{L·atm/(K·mol)}) \times (273 \, \text{K})} \]
\[ n \approx \frac{4}{22.414} \approx 0.178 \, \text{mol} \]

### Step 4: Calculate the mass of CO₂ needed
The molar mass of CO₂ is:
\[ \text{Molar mass of CO₂} = 12 + (2 \times 16) = 44 \, \text{g/mol} \]

Now, calculate the mass (\(W\)):
\[ W = n \times \text{Molar mass} = 0.178 \, \text{mol} \times 44 \, \text{g/mol} \approx 7.832 \, \text{g} \]

### Step 5: Calculate the volume of liquid CO₂ needed using its density
Using the formula for density:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \implies \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]
Substituting the values:
\[ \text{Volume} = \frac{7.832 \, \text{g}}{0.8 \, \text{g/cm}^3} \approx 9.79 \, \text{cm}^3 \]

### Conclusion
The volume of the cartridge of liquid CO₂ required to inflate the life jacket is approximately \(9.79 \, \text{cm}^3\).

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