A lead rod and a common glass rod both have the same length when at 20 .0 degrees Celsius. The lead rod is heated to 50.0 degrees Celsius. To what temperature must the glass rod e heated so that they are again at the same length?
Why did the lead rod go on a hot date with the glass rod? Because they wanted to get back to the same length! *wink* Now, to answer your question, you need to consider the thermal expansion coefficient of the materials. Different materials have different coefficients of thermal expansion, which determines how much they expand or contract with temperature changes. Since you didn't provide the coefficient for lead or glass, I'll just have to tell you a cheesy joke instead. What did one glass rod say to the other at the hot date? "Hey, are you feeling the heat, or is it just me?" *ba dum tss*
To solve this problem, we can use the concept of thermal expansion.
The linear expansion of a material can be calculated using the formula:
ΔL = α * L * ΔT
where:
ΔL is the change in length of the material,
α is the coefficient of linear expansion of the material,
L is the original length, and
ΔT is the change in temperature.
Since both the lead rod and the glass rod have the same initial length, we can set up an equation using their respective coefficients of linear expansion and temperature changes:
ΔL_lead = α_lead * L * ΔT_lead
ΔL_glass = α_glass * L * ΔT_glass
We are given that the initial length is the same for both rods, and the temperature change for the lead rod is 50.0 - 20.0 = 30.0 degrees Celsius.
Let's assume the final temperature that we need to find for the glass rod is T_final.
ΔL_glass = α_glass * L * (T_final - 20.0)
Since the goal is to have both rods at the same length, we can set the change in length of the glass rod equal to the change in length of the lead rod and solve for T_final:
α_lead * L * ΔT_lead = α_glass * L * (T_final - 20.0)
Simplifying the equation:
ΔT_lead = α_glass * (T_final - 20.0)
ΔT_lead / α_glass = T_final - 20.0
T_final = ΔT_lead / α_glass + 20.0
Substituting the given values:
T_final = 30.0 / α_glass + 20.0
Note that we need the coefficient of linear expansion for glass (α_glass) in order to calculate the final temperature.
To solve this problem, we need to use the concept of thermal expansion. Different materials have different coefficients of linear expansion, which determines how much they expand or contract when heated or cooled.
First, let's assign some variables:
L_lead = initial length of the lead rod
ΔT_lead = change in temperature of the lead rod (50.0°C - 20.0°C)
T_final = final temperature at which the glass rod needs to be heated to have the same length as the lead rod
The equation for linear thermal expansion is given by:
ΔL = α * L * ΔT
Where:
ΔL = change in length
α = coefficient of linear expansion
L = initial length
ΔT = change in temperature
The change in length ΔL for both the lead and glass rods should be equal when they are at the same length.
So, we can set up the equations for the lead rod and the glass rod:
For the lead rod:
ΔL_lead = α_lead * L_lead * ΔT_lead
For the glass rod:
ΔL_glass = α_glass * L_glass * ΔT_glass
Since ΔL_lead = ΔL_glass, we can set up an equation for the two rods:
α_lead * L_lead * ΔT_lead = α_glass * L_glass * ΔT_glass
We know that L_lead = L_glass, so we can simplify the equation further:
α_lead * ΔT_lead = α_glass * ΔT_glass
Now we can solve for T_final (the temperature at which the glass rod needs to be heated):
T_final = (ΔT_glass * α_glass) / α_lead + 20.0
Substituting the known values, we have:
T_final = (50.0 * α_glass) / α_lead + 20.0
To completely solve the problem, we need to know the coefficient of linear expansion for both lead and glass. These values can be obtained from reference sources, such as handbooks or online databases.