The density of a solid is 130gcm^-3 at the temperature of 25 degree Centrigate.Find the Density at 150 degree Celcius if the linear expansion of the solid is 2.0*10^(-5)K^-1. I got the answer to be 129.03.
To find the density of the solid at 150 degrees Celsius, we can use the formula for volumetric expansion:
ΔV = V₀ * β * ΔT
Where:
ΔV is the change in volume
V₀ is the initial volume
β is the coefficient of volumetric expansion
ΔT is the change in temperature
Since we are given the linear expansion coefficient (α), we can relate it to the volumetric expansion coefficient (β) using the equation:
β = 3α
Now, let's calculate the change in volume:
ΔV = V₀ * β * ΔT
ΔV = (130 cm³) * (3 * 2.0 * 10^(-5) K^(-1)) * (150 - 25) °C
ΔV = (130 cm³) * (6.0 * 10^(-5) K^(-1)) * (125) °C
ΔV = 0.0975 cm³
Next, calculate the final volume:
V = V₀ + ΔV
V = 130 cm³ + 0.0975 cm³
V = 130.0975 cm³
Finally, calculate the density at 150 degrees Celsius:
ρ = m / V
ρ = (130 g) / (130.0975 cm³)
ρ ≈ 0.99962 g/cm³
So, the density of the solid at 150 degrees Celsius is approximately 0.99962 g/cm³, which is approximately equal to 130.03 (rounded to two decimal places). Therefore, your answer of 129.03 is not correct.
To find the density at a higher temperature, you need to consider the change in volume due to the expansion of the solid. The formula for linear expansion is given by:
ΔL = α * L * ΔT
Where:
ΔL is the change in length
α is the linear expansion coefficient
L is the original length
ΔT is the change in temperature
The change in volume (ΔV) due to linear expansion is given by:
ΔV = β * V * ΔT
Where:
β is the volume expansion coefficient (which is three times the linear expansion coefficient)
V is the original volume
ΔT is the change in temperature
Now, let's calculate the change in volume. Since density (ρ) is defined as the mass (m) divided by volume (V), we can rewrite it as:
ρ = m / (V + ΔV)
We know that the density at 25 degrees Celsius is 130 g/cm^3. Given the linear expansion coefficient (α = 2.0 * 10^(-5) K^(-1)) and the temperature change (ΔT = 150 - 25 = 125 degrees Celsius), we can calculate the change in volume using the formula mentioned above.
Since the volume expansion coefficient (β) is three times the linear expansion coefficient (α), we have β = 3 * α. Here, β = 2.0 * 10^(-5) K^(-1) * 3 = 6.0 * 10^(-5) K^(-1).
Let's say the original volume is V. The change in volume (ΔV) is then given by:
ΔV = β * V * ΔT = (6.0 * 10^(-5) K^(-1)) * V * 125
Now we can substitute the value of ΔV into the density formula:
ρ = m / (V + ΔV) = m / (V + (6.0 * 10^(-5) K^(-1)) * V * 125)
Since we know the density at 25 degrees Celsius and the formula ρ = m / V, we can set up the following equation:
130 g/cm^3 = m / V
Solving the equation for m, we get:
m = 130 g/cm^3 * V
Substituting this value in the density formula:
ρ = (130 g/cm^3 * V) / (V + (6.0 * 10^(-5) K^(-1)) * V * 125)
Now we can substitute the value of V with previous calculations or a given condition to find the density at 150 degrees Celsius. Plugging in the values and performing the calculation will yield the final answer.