Deriving Specific Heat Equations

[jasmine689]

Homework Statement

Derive the Following Equations:
c = [(m_w*c_w + m_c*c_al) (T_e - T_i)] / m_s*(T_s - T_e)
and
L = (m_w*c_w + m_c*c_al) * (T_i - T_e) - m_I*c_w * [(T_e- T_m)/m_I]

Homework Equations

T_i - initial temp of water and calorimeter
T_m - melting temp of ice
T_e - equilibrium temp of system
m_w- mass of water
m_c - mass of calorimeter
m_I - mass of ice
c_w - specific heat of water
c_al - specifical heat of aluminum
L - unknown heat of fusion

The Attempt at a Solution

...trying to receive help from my teacher's assistant, but not being able to do so :[

Any attempts at either equations, wrong or right, will be fully appreciated.

 

[jasmine689]

PLEASE ANY kind of help, even if it's not completed, would be REALLY appreciated :)

...I need this done in by 3pm today...

 

[nlightner]

The specific heat equation is a fundamental equation used in thermodynamics to calculate the amount of heat energy needed to raise the temperature of a substance. It is derived from the first law of thermodynamics, which states that energy cannot be created or destroyed, only transferred from one form to another.

To derive the first equation, we can start by considering the heat energy transferred from the water and calorimeter to the system, which is equal to the heat energy gained by the system. This can be written as:

Q = (m_w * c_w + m_c * c_al) * (T_e - T_i)

Where Q is the heat energy, m_w and m_c are the masses of water and calorimeter, c_w and c_al are the specific heats of water and aluminum, and T_e and T_i are the initial and equilibrium temperatures of the system.

Next, we can consider the heat energy lost by the system due to the change in temperature of the system (from T_s to T_e). This can be written as:

Q = m_s * c_s * (T_s - T_e)

Where m_s is the mass of the system and c_s is its specific heat.

Equating these two equations, we get:

(m_w * c_w + m_c * c_al) * (T_e - T_i) = m_s * c_s * (T_s - T_e)

Solving for c_s, we get:

c_s = [(m_w * c_w + m_c * c_al) * (T_e - T_i)] / (m_s * (T_s - T_e))

This is the first equation that we were asked to derive.

To derive the second equation, we can use the same approach. We can consider the heat energy gained by the water and calorimeter, which is equal to the heat energy lost by the ice and system. This can be written as:

Q = (m_w * c_w + m_c * c_al) * (T_i - T_e) + m_I * c_w * (T_e - T_m)

Where m_I is the mass of ice and T_m is the melting temperature of ice.

Next, we can consider the heat energy gained by the system due to the heat of fusion (L) of the ice. This can be written as:

Q = m_I * L

Equating these two equations, we get:

(m_w * c_w + m_c *