Question

Toricelli’s barometer used mercury. Pascal duplicated it using French wine of density 984 kg m^–3. Determine the height of the wine column for normal atmospheric pressure.

Answer 1

Principle of Barometer

Atmospheric pressure supports liquid column:

$$P_\text{atm} = \rho g h$$ $$h = \frac{P_\text{atm}}{\rho g}$$

Given Data

• Wine density: \(\rho_\text{wine} = 984\) kg/m³
• Mercury: \(\rho_\text{Hg} = 13{,}600\) kg/m³ (standard 76 cm)
• \(P_\text{atm} = 1.013 \times 10^5\) Pa (or use mercury equivalent)
• \(g = 9.8\) m/s²

Calculation Methods

Mercury (Reference)

\(h_\text{Hg} = 0.76\) m

Supports \(P_\text{atm}\)

Wine (Pascal)

\(h_\text{wine} = ?\)

Same \(P_\text{atm}\)

Method 1: Density ratio (simplest)

$$\frac{h_\text{wine}}{h_\text{Hg}} = \frac{\rho_\text{Hg}}{\rho_\text{wine}}$$ $$h_\text{wine} = 0.76 \times \frac{13{,}600}{984} = 0.76 \times 13.82 = 10.50 \, \text{m}$$

Method 2: Direct pressure

$$h_\text{wine} = \frac{1.013 \times 10^5}{984 \times 9.8} = \frac{1.013 \times 10^5}{9643} \approx 10.51 \, \text{m}$$

Wine Column Height

\(h = \textbf{10.5 m}\)

Verification

• Ratio check: \(\frac{13600}{984} \approx 13.8\), \(0.76 \times 13.8 \approx 10.5\) ✓
• Practical: Wine barometer needs ~10.5 m tube (Pascal's experiment) ✓

Historical Note

Pascal used red wine, observed ~10 m column, confirming Torricelli's vacuum above liquid.