The bulk modulus of water is B = 2.2 x 10^9 N/m2. What change in pressure Delta P (in atmospheres) is required to keep water from expanding when it is heated from 12.4 °C to 31.1 °C?
I know that the equation for pressure is Delta P = Force / Area but I have no idea what I have to do before this step.
Please help.
To solve this problem, you need to apply the concept of bulk modulus, which relates the change in pressure to the change in volume of a substance.
The formula for bulk modulus (B) is:
B = -(Delta P / Delta V / V)
where ΔP is the change in pressure, ΔV is the change in volume, and V is the initial volume.
In this case, you want to prevent water from expanding, which means the change in volume (ΔV) should be zero. Therefore, the formula can be simplified to:
B = -(ΔP / V)
Now, you need to rearrange the formula to solve for ΔP:
ΔP = -B * V
Given that the bulk modulus of water (B) is 2.2 x 10^9 N/m^2 and the initial volume (V) is the volume of water, we can calculate ΔP. But first, we need to convert the temperature change from Celsius to Kelvin, as the Kelvin scale is used in thermodynamic calculations.
To convert the temperature from Celsius to Kelvin, you use the formula:
T(K) = T(°C) + 273.15
So, the initial temperature (T1) in Kelvin would be:
T1(K) = 12.4 + 273.15
And the final temperature (T2) in Kelvin would be:
T2(K) = 31.1 + 273.15
Now, let's calculate the volume change (ΔV). For most liquids, the change in volume due to temperature change can be approximated by the formula:
ΔV = beta * V0 * ΔT
where beta is the average coefficient of volumetric expansion, V0 is the initial volume, and ΔT is the temperature change in Kelvin.
The average coefficient of volumetric expansion for water is approximately 0.000207 K^(-1).
Substituting the given values into the formula:
ΔV = 0.000207 * V * (T2(K) - T1(K))
Now you have the values necessary to calculate ΔP:
ΔP = -B * V
where B is the bulk modulus of water and V is the volume of water.
By substituting the given values into the equation, you can find the change in pressure (ΔP) in N/m^2.
To convert the pressure from N/m^2 to atmospheres, you can use the conversion factor:
1 atmosphere = 1.01325 × 10^5 N/m^2.
So, you can divide the calculated ΔP in N/m^2 by 1.01325 × 10^5 to obtain the pressure change in atmospheres.