Q5.15P - Chemistry: Molecular Nature Of Matter And Change

Question

The gravitational force exerted by an object is given by F = mg, where F is the force in Newtons, m is the mass in kilograms, and g is the acceleration due to gravity (9.81 m/s²).

(a) Use the definition of the pascal to calculate the mass (in kg) of the atmosphere above 1 m² of ocean.

(b) Osmium (Z = 76) is a transition metal in Group 8B(8) and has the highest density of any element (22.6 g/mL). If an osmium column is 1 m² in area, how high must it be for its pressure to equal atmospheric pressure? [Use the answer from part (a) in your calculation.]

Step-by-Step Solution

Answer

1. The mass of the atmosphere is $1.09 \times {10}^4 \text{kg}$.
2. The height of the column is $0.457 \text{m}$ .

Step 1: Determination of mass

1 Pascal is the pressure exerted on a surface when a force of 1 Newton acts on a surface area of 1m².

 $\text{Pressure} = \frac{\text{Force}}{\text{Area}}$

$1 \text{Pa} = \frac{\text{1} \text{N}}{\text{1} {\text{m}}^{\text{2}}}$

 According to the given data, F = mg

Where, 

F = force

m = mass

g = acceleration due to gravity

 1 atm = $1.01325 \times {10}^{5 }\text{Pa}$

 Now, the mass of the atmosphere is,

 $\begin{array}{rcl}1.01325 \times {10}^5 \text{Pa} & =& \frac{\text{mg}}{\text{area}}\\ 1.01325 \times {10}^5 \text{Pa} & =& \frac{\text{m}\times \text{9.8} {\text{m/s}}^{\text{2}}}{\text{1} {\text{m}}^{\text{2}}}\\ \text{m} & =& 1.09 \times {10}^4\text{kg}\\ & & \end{array}$

Thus, the mass of the atmosphere is $1.09 \times {10}^4 \text{kg}$.

Step 2: Determination of height of the column

The relation between density and pressure is,

 $P=pgh$

Where,

 data-custom-editor="chemistry" $$\begin{gathered} \rho = \text{density} \\ \text{g} =\text{acceleration} \text{due} \text{to} \text{gravity} \\ \text{h} = \text{height} \end{gathered}$$

 Now, the height of the column is,

 data-custom-editor="chemistry" $$\begin{gathered} \mathrm{h}=\frac{\mathrm{P}}{\mathrm{pg}}=\frac{1.01325\times {10}^5 \text{Pa}}{22.6 \text{g/ml×}9.81 {\text{m/s}}^{\text{2}}} \\ \text{h} = 0.457 \text{m} \end{gathered}$$

Thus, the height of the column is $0.457 \text{m}$