# Capillary Rise/Drop in a Tube: Glycerin Example Let's solve the problem step by step: ## **Step 1: Identify Relevant Formula** For a liquid in a capillary tube: - If the liquid wets the tube (like water or glycerin in glass), there is a **rise**. - If not (like mercury in glass), there is a **depression**. The capillary rise \( h \) is given by: \[ h = \frac{4T \cos \theta}{\rho g d} \] where: - \( T \) = surface tension (N/m) - \( \theta \) = contact angle (for glycerin on glass, assume \( \theta \approx ^\circ \), so \( \cos \theta = 1 \)) - \( \rho \) = density of liquid (kg/m³) - \( g \) = acceleration due to gravity (\(9.81 \) m/s²) - \( d \) = diameter of tube (m) ## **Step 2: List Given Data** - **Tube diameter** \( d = 5.\,\text{mm} = .005\,\text{m} \) - **Height of liquid** (initially present, not directly needed) - **Surface tension** \( T = .32\,\text{N/m} \) - **Specific gravity of glycerin** \( S = 1.25 \) - **Density of water** \( \rho_{\text{water}} = 100\,\text{kg/m}^3 \) - So, \( \rho = S \times \rho_{\text{water}} = 1.25 \times 100 = 125\,\text{kg/m}^3 \) - **Contact angle**: For glycerin on glass, \( \theta \approx ^\circ \). ## **Step 3: Apply Formula** Plug in the values (with \( \cos ^\circ = 1 \)): \[ h = \frac{4 \times .32 \times 1}{125 \times 9.81 \times .005} \] ## **Step 4: Calculate Numerator and Denominator** - Numerator: \( 4 \times .32 = 1.28 \) - Denominator: \( 125 \times 9.81 \times .005 \) - \( 9.81 \times .005 = .04905 \) - \( 125 \times .04905 = 61.3125 \) ## **Step 5: Calculate Rise** \[ h = \frac{1.28}{61.3125} \approx .02088\,\text{m} = 20.88\,\text{mm} \] ## **Step 6: Find Change in Height** - **Original height**: \( 20.\,\text{mm} \) - **Capillary rise**: \( 20.88\,\text{mm} \) - **New height**: \( 20. + 20.88 = 40.88\,\text{mm} \) - **Rise due to capillarity**: \( 20.88\,\text{mm} \) ## **Final Answer** **The rise in the height of glycerin in the tube due to capillarity is \(\boxed{20.9\,\text{mm}}\) (rounded to one decimal place).** --- ### **Summary Table** | Quantity | Value | |----------------------|---------------------------| | Tube diameter (\(d\))| \(.005\,\text{m}\) | | Surface tension (\(T\))| \(.32\,\text{N/m}\) | | Specific gravity (\(S\))| \(1.25\) | | Density (\(\rho\)) | \(125\,\text{kg/m}^3\) | | Capillary rise (\(h\))| \(20.9\,\text{mm}\) |