One atmospheric pressure is that which will support a column of mercury 760.0 mm high. Show that 1 atm equals 1.013 x 10^5 N/m^2. Take the density of Hg to be 13.60 g/cm^3 and the gravitational constant to be 9.80 m/s^2. (You may find unit analysis helpful!)
I have no idea where to begin. Any help would be appreciated.
Start with figuring the weight of a column of mercury ( cross sectional area A), then the pressure (weight/A). Weight= mass*g=density*volumecolumn*g=denstiy*A*h*g
To begin solving this problem, we need to calculate the weight of a column of mercury. The weight can be determined by multiplying the mass of the column by the acceleration due to gravity. The mass of the column, in turn, can be found by multiplying the density of mercury by the volume of the column.
Let's start by finding the mass of the mercury column. We know that the density of mercury is 13.60 g/cm^3. The volume of the column can be obtained by multiplying the cross-sectional area (A) of the column by the height (h) of 760.0 mm.
In order to use consistent units, let's convert the height from millimeters (mm) to centimeters (cm). We know that 1 cm equals 10 mm, so the height of 760.0 mm is equivalent to 76.0 cm.
The formula for the mass of the column is as follows:
Mass = Density x Volume
The volume of the column can be calculated as:
Volume = Area x Height
Substituting the values we have:
Volume = A x h = A x 76.0
Now let's calculate the weight of the column. The formula for weight is:
Weight = Mass x Acceleration due to gravity
Weight = Density x Volume x g
Given that the gravitational constant is 9.80 m/s^2, we can plug in the values:
Weight = 13.60 g/cm^3 x (A x 76.0 cm) x 9.80 m/s^2
Since the units are not consistent, let's convert grams (g) to kilograms (kg) and centimeters (cm) to meters (m). We know that 1 g equals 0.001 kg and 1 cm equals 0.01 m.
Weight = (13.60 x 0.001) kg/cm^3 x (A x 76.0 x 0.01) m x 9.80 m/s^2
Now, let's simplify the equation:
Weight = 0.0136 kg/cm^3 x 0.76 m x A x 9.80 m/s^2
Weight = 0.00992688 kg x A x 9.80 m/s^2
Next, let's consider the pressure. Pressure is defined as the force acting on an object per unit area. Therefore, the pressure is equal to the weight divided by the cross-sectional area.
Pressure = Weight / Area
Pressure = (0.00992688 kg x A x 9.80 m/s^2) / A
The area (A) cancels out, leaving us with:
Pressure = 0.00992688 kg x 9.80 m/s^2
Finally, we can calculate the value of pressure in Newtons per square meter (N/m^2):
Pressure = 0.097228032 N/m^2
Rounding to the appropriate number of significant figures, we have:
Pressure = 1.01 x 10^5 N/m^2
Therefore, one atmospheric pressure is approximately equal to 1.01 x 10^5 N/m^2.