what should be the length of aluminum and a brass rods at STP. so that brass rod is longer than the aluminum rod by 4 cm at all conditions of temperature

Aluminum's thermal coefficient of expansion is about 25% higher than that of brass. See

http://hyperphysics.phy-astr.gsu.edu/hbase/tables/thexp.html
for verification.

Let L1' = Al rod length at 0 C, in cm
Let L2' = Brass rod length at 0 C, in cm
T = rod temperature in deg. C
a = thermal expansion coefficient of brass.

L1' - L2' = 4
At any temperature T,
L1' + 1.25*a*L1*(T-0) - L2' - a*L2(T-0)
= L1' - L2' -(0.25*a*L1*T - L2*a*T)
= 4 at any T, if 0.25 L1 = L2

To determine the required lengths of the aluminum and brass rods at STP (Standard Temperature and Pressure), considering a 4 cm difference in length at all conditions of temperature, we need to use the concept of thermal expansion.

The formula for linear thermal expansion is given by:

ΔL = α * L0 * ΔT

Where:
ΔL is the change in length
α is the coefficient of linear expansion
L0 is the original length
ΔT is the change in temperature

Since we want to maintain a constant difference in length between the aluminum and brass rods at all temperatures, we can set up the following equation:

Lbrass - Laluminum = 4 cm

Now, let's calculate the lengths of the rods at STP.

At STP, the temperature is 0°C or 273 K.

First, we need to find the coefficient of linear expansion for aluminum and brass.

The coefficient of linear expansion for aluminum is αaluminum = 0.000022 / °C or 22 x 10^-6 / °C.
The coefficient of linear expansion for brass is αbrass = 0.000019 / °C or 19 x 10^-6 / °C.

Now, we can use the formula for thermal expansion to calculate the lengths of the rods at STP.

Let's assume the initial length of the aluminum rod is Laluminum.

For the brass rod, its length would be Laluminum + 4 cm.

Setting up the equation:

Lbrass - Laluminum = 4 cm

Using the formula for thermal expansion for each material:

(αbrass * Laluminum * ΔT) - (αaluminum * Laluminum * ΔT) = 4 cm

Simplifying the equation:

(αbrass - αaluminum) * Laluminum * ΔT = 4 cm

Substituting the values:

(19 x 10^-6 / °C - 22 x 10^-6 / °C) * Laluminum * ΔT = 4 cm

Solving for Laluminum:

Laluminum = (4 cm) / ((19 x 10^-6 / °C - 22 x 10^-6 / °C) * ΔT)

By substituting ΔT with 273 K for STP conditions:

Laluminum = (4 cm) / ((19 x 10^-6 / °C - 22 x 10^-6 / °C) * 273 K)

Calculating the value, we can find the required length of the aluminum rod at STP to maintain a 4 cm difference in length with the brass rod at all temperatures.