91Ó°ÊÓ

[II] A glass stopper has a mass of \(2.50 \mathrm{~g}\) when measured in air, \(1.50\) g in water, and \(0.70 \mathrm{~g}\) in sulfuric acid. What is the density of the acid? What is its specific gravity? The \(F_{B}\) on the stopper in water is \((0.00250-0.00150)(9.81) \mathrm{N}\). This is the weight of the displaced water. Since \(\rho=m / V\), or \(\rho g=\) \(F_{W} / V\), $$ \begin{array}{l} \text { Volume of stopper }=\text { Volume of displaced water }=\frac{\text { weight }}{\rho \mathrm{g}} \\ V=\frac{(0.00100)(9.81) \mathrm{N}}{\left(1000 \mathrm{~kg} / \mathrm{m}^{3}\right)\left(9.81 \mathrm{~m} / \mathrm{s}^{2}\right)}=1.00 \times 10^{-6} \mathrm{~m}^{3} \end{array} $$ The buoyant force in acid is $$ \left[(2.50-0.70) \times 10^{-3}\right](9.81) \mathrm{N}=(0.00180)(9.81) \mathrm{N} $$ But this is equal to the weight of displaced acid, \(m g\). Since \(\rho=\) \(m / V\), and since \(m=0.00180 \mathrm{~kg}\) and \(V=1.00 \times 10^{-6} \mathrm{~m}^{3}\), $$ \rho \text { of acid }=\frac{0.00180 \mathrm{~kg}}{1.00 \times 10^{-6} \mathrm{~m}^{3}}=1.8 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3} $$ Then, for the acid, $$ \text { sp } \mathrm{gr}=\frac{\rho \text { of acid }}{\rho \text { of water }}=\frac{1800}{1000}=1.8 $$ Then, since sp gr of acid \(=(\rho\) of acid \() /(\rho\) of water), \(\rho\) of acid \(=(\) sp gr of acid \()(\rho\) of water \()=(1.8)\left(1000 \mathrm{~kg} / \mathrm{m}^{3}\right)=1.8 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\)

Step by step solution

01

Calculate Buoyant Force in Water

The buoyant force \(F_B\) in water is the weight of the stopper displaced in water. Given the stopper mass difference in air and water, we have \((2.50 - 1.50)\, \text{g} = 1.00\, \text{g} = 0.00100\, \text{kg}\). The force is calculated as \(F_B = 0.00100 \times 9.81\, \text{N} = 0.00981\, \text{N}\).

02

Determine Volume of Stopper

The volume \(V\) of the stopper is equal to the volume of the displaced water. Using the formula \(F_B = \rho_{\text{water}} \cdot V \cdot g\), we find \(V = \frac{0.00981\, \text{N}}{1000\, \text{kg/m}^3 \times 9.81\, \text{m/s}^2} = 1.00 \times 10^{-6}\, \text{m}^3\).

03

Compute Buoyant Force in Acid

For the acid, the buoyant force is \((2.50 - 0.70)\, \text{g} = 1.80\, \text{g} = 0.00180\, \text{kg}\). Thus, \(F_B = 0.00180 \times 9.81\, \text{N} = 0.017658\, \text{N}\).

04

Calculate Density of the Acid

The density \(\rho\) of the acid is calculated using \(\rho = \frac{m}{V}\), where the mass \(m = 0.00180\, \text{kg}\) and volume \(V = 1.00 \times 10^{-6}\, \text{m}^3\). Thus, \(\rho = \frac{0.00180}{1.00 \times 10^{-6}} = 1800\, \text{kg/m}^3\).

05

Determine Specific Gravity of the Acid

The specific gravity (sp gr) of the acid is the ratio of the density of the acid to the density of water. Hence, sp gr = \(\frac{1800\, \text{kg/m}^3}{1000\, \text{kg/m}^3} = 1.8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Buoyant Force

The buoyant force is an upward force exerted by a fluid on an object placed in it. This force allows objects to float or rise in fluids. It is equal to the weight of the fluid displaced by the object. This concept is based on Archimedes' principle, which states that an object submerged in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.

To calculate the buoyant force, you must find the difference in weight of the object when it is in air versus when it is submerged in the fluid. For example, in water, a glass stopper weighed 2.50 g in air and 1.50 g in water. The difference (1.00 g) represents the mass of water displaced by the stopper.

- Convert the mass difference to kilograms by multiplying by 0.001. So, 1.00 g becomes 0.001 kg. - Multiply this mass by the acceleration due to gravity (9.81 m/s²) to find the buoyant force, which is 0.00981 N in water. Buoyant force helps in understanding the floating and sinking behaviors of objects and is crucial in calculating an object's density when immersed in various fluids.

Specific Gravity

Specific gravity is a dimensionless number that compares the density of a substance to the density of a reference substance, usually water for liquids and solids. It shows how heavy or light a substance is compared to water.Specific gravity is calculated by dividing the density of the object by the density of the reference fluid (water). Here, the density of the sulfuric acid was determined to be 1800 kg/m³. The density of water is typically about 1000 kg/m³.

- Specific Gravity (sp gr) = \(\frac{\text{Density of substance}}{\text{Density of water}}\).- In this case, \(\frac{1800}{1000} = 1.8\).This means that sulfuric acid is 1.8 times denser than water. Unlike density, specific gravity has no units and provides a way to quickly assess whether substances will sink or float when mixed with water.

This concept is vital in various fields, such as geology, material science, and chemical engineering, as it helps characterize substances based on their composition and impurity levels.

Volume of Displacement

Volume of displacement refers to the volume of fluid displaced by an object when it is placed in a fluid. This concept is key in calculating the buoyant force and involves understanding the object's geometry and its submerged depth.
To find the volume of displacement:- Measure or calculate the buoyant force (weight of the displaced fluid).- Use the relation \(F_B = \rho \cdot V \cdot g\), where \(F_B\) is the buoyant force, \(\rho\) is the fluid's density, \(V\) is the volume displacement, and \(g\) is the gravitational acceleration (9.81 m/s²).- Rearrange and solve for \(V\). For the glass stopper example, using the buoyant force in water (0.00981 N) and the density of water (1000 kg/m³), the volume is \(\frac{0.00981}{1000 \times 9.81} = 1.00 \times 10^{-6} \text{ m}^3\).Understanding the volume of displacement helps determine how much fluid is pushed aside by an object and is crucial in designing ships, submarines, and calculating loads on bridges in flood conditions.