Answer:
The height is 0.247 m.
Explanation:
Given that,
Spring constant = 1.60\times10^3\ N/m[/tex]
Pressure = 1.00 atm
Temperature = 20.0°C
Suppose if the piston has a cross section area of 0.0120 m² and negligible mass, how high will it rise when the temperature is raised to 250°C
The equation of the pressure is
[tex]P'=P_{0}+\dfrac{kh}{A}[/tex]...(I)
The equation of the volume is
[tex]V'=V_{0}+Ah[/tex]....(II)
We need to calculate the height
Using equation of ideal gas
[tex]\dfrac{P_{0}V}{T}=\dfrac{P'V'}{T'}[/tex]
[tex]P'V'=P_{0}V\dfrac{T'}{T}[/tex]
Put the value of P' and V' from equation (I) and (II)
[tex]P_{0}+\dfrac{kh}{A}\times V_{0}+Ah=P_{0}V\dfrac{T'}{T}[/tex]
Put the value into the formula
[tex]1.013\times10^{5}+\dfrac{1.60\times10^{3}\times h}{0.0120 }\times5.00\times10^{-3}+0.0120 \times h=1.013\times10^{5}\times5.00\times10^{-3}\times\dfrac{523}{293}[/tex]
[tex]1.013\times10^{5}\times5.00\times10^{-3}+1.013\times10^{5}\times0.0120\times h+1600h^2=904.09[/tex]
[tex]1600h^2+1215.6h=904.09-506.5[/tex]
[tex]1600h^2+1215.6h=397.59[/tex]
[tex]h=0.247\ m[/tex]
Hence, The height is 0.247 m.
