A density battle holds 250 g of liquid at 30°C ar only 248.5 g at 60°C. Find
a. the apparent and
b. the real cubic expansivity of the liquid, if ti linear expansivity of the material of the bott is a = 0.000006 K
To solve this problem, we can use the formula for apparent density:
ρ_apparent = (m_initial - m_final) / (V_initial - V_final)
where ρ_apparent is the apparent density, m_initial and m_final are the initial and final masses of the liquid, V_initial and V_final are the initial and final volumes of the liquid.
a. To find the apparent density, we need to convert the given masses into volumes using the densities at the given temperatures. We can use the formula for density:
ρ = m / V
where ρ is the density, m is the mass, and V is the volume.
At 30°C:
ρ_initial = m_initial / V_initial
At 60°C:
ρ_final = m_final / V_final
To eliminate the volume terms, we can divide the equations:
ρ_initial / ρ_final = (m_initial / V_initial) / (m_final / V_final)
Simplifying, we get:
ρ_final = (ρ_initial * V_final) / V_initial
We know that the density is inversely proportional to the volume, so:
ρ_final = (ρ_initial * V_final) / V_initial = ρ_initial * (V_final / V_initial)
Now we can use the formula for apparent density:
ρ_apparent = (m_initial - m_final) / (V_initial - V_final) = (ρ_initial - ρ_final) / (V_initial - V_final)
Substituting in the values we know:
ρ_apparent = (ρ_initial - ρ_initial * (V_final / V_initial)) / (V_initial - V_final)
Simplifying further:
ρ_apparent = ρ_initial * (1 - (V_final / V_initial)) / (V_initial - V_final)
We can use the equation m = ρ * V to rewrite this equation:
ρ_apparent = (m_initial / V_initial) * (1 - (m_final / m_initial)) / ((m_initial / ρ_initial) - (m_final / ρ_initial))
Simplifying further:
ρ_apparent = (m_initial - m_final) / ((m_initial / ρ_initial) - (m_final / ρ_initial))
Substituting in the given values:
ρ_apparent = (250 g - 248.5 g) / ((250 g / ρ_initial) - (248.5 g / ρ_initial))
We can solve for ρ_initial by rearranging the equation for density:
ρ_initial = m_initial / V_initial
Using the values we know (m_initial = 250 g and V_initial = m_initial / ρ_initial), we get:
ρ_initial = 250 g / (m_initial / ρ_initial) = (250 g * ρ_initial) / m_initial
Substituting this into the equation for ρ_apparent:
ρ_apparent = (250 g - 248.5 g) / ((250 g * ρ_initial) / m_initial - (248.5 g * ρ_initial) / m_initial)
Simplifying further:
ρ_apparent = (250 g - 248.5 g) / ((250 g - 248.5 g) * ρ_initial / m_initial)
ρ_apparent = m_initial / m_initial
ρ_apparent = 1
Therefore, the apparent density of the liquid is 1.
b. The real cubic expansivity (α) of the liquid can be calculated using the formula:
α = (ρ_final - ρ_initial) / (ρ_initial * ΔT)
where α is the real cubic expansivity, ρ_final and ρ_initial are the final and initial densities of the liquid, and ΔT is the change in temperature.
Substituting in the given values:
α = (ρ_final - ρ_initial) / (ρ_initial * ΔT)
To find ρ_final, we can use the formula for density:
ρ_final = m_final / V_final
Substituting the given values:
ρ_final = 248.5 g / V_final
Using the linear expansivity equation:
V_final = V_initial * (1 + α * ΔT)
Substituting the given values:
V_final = V_initial * (1 + 0.000006 K * (60°C - 30°C))
V_final = V_initial * (1 + 0.000006 K * 30 K)
V_final = V_initial * (1 + 0.00018)
V_final = 1.00018 * V_initial
Now we can substitute this into the equation for ρ_final:
ρ_final = 248.5 g / (1.00018 * V_initial)
ρ_final = 248.3 g / V_initial
Substituting these values into the equation for α:
α = ((248.3 g / V_initial) - 1) / (1 * (60°C - 30°C))
Simplifying:
α = (248.3 g / V_initial - 1) / 30 K
Therefore, the real cubic expansivity of the liquid is (248.3 g / V_initial - 1) / 30 K.