A coin dealer, offered a rare silver coin, suspected that it might be a counterfeit nickel copy. The dealer heated the coin, which weighed 14.0 g to 100°C in boiling water and then dropped the hot coin into 29.5 g of water at T = 15.0°C in an insulated coffee-cup, and measured the rise in temperature. If the coin was really made of silver, what would the final temperature of the water be (in °C)? (for nickel, s = 0.445 J/g°C; for silver, s = 0.233 J/g°C )

Well, if the coin is a counterfeit nickel copy, maybe it's trying to "nickel and dime" you. But let's not jump to conclusions just yet.

To solve this problem, we can use the principle of heat transfer. The heat lost by the hot coin will be gained by the cold water. Assuming no heat is lost to the surroundings, we can set up an equation:

q_hot = q_cold

The heat lost by the hot coin can be calculated using the equation:

q_hot = m_hot * s_hot * ΔT

where m_hot is the mass of the coin, s_hot is the specific heat capacity of silver, and ΔT is the change in temperature.

Similarly, the heat gained by the cold water can be calculated using the equation:

q_cold = m_cold * s_cold * ΔT

where m_cold is the mass of the water, s_cold is the specific heat capacity of water, and ΔT is the change in temperature.

Since the heat lost by the hot coin is equal to the heat gained by the cold water, we have:

m_hot * s_hot * ΔT = m_cold * s_cold * ΔT

Now, let's plug in the given values:

(14.0 g) * (0.233 J/g°C) * (100°C - T) = (29.5 g) * (4.18 J/g°C) * (T - 15.0°C)

Solving this equation will give us the final temperature of the water (T).

Aaaand, I apologize. I just realized that my previous remark about the coin being a "nickel copy" wasn't really funny. I guess that joke was just a coin-flip. But hey, let's get back to the problem!

To solve this problem, we need to use the principle of heat transfer. The heat lost by the coin will be equal to the heat gained by the water.

The amount of heat transferred (Q) can be calculated using the formula:

Q = m * c * ΔT

where:
Q = heat transferred (in Joules)
m = mass (in grams)
c = specific heat capacity (in J/g°C)
ΔT = change in temperature (final temperature - initial temperature)

Let's calculate the amount of heat lost by the coin. The coin initially had a temperature of 100°C and was heated to 100°C. Therefore, the change in temperature (ΔT) for the coin is 0°C.

For the water, the mass (m) is 29.5 g and the initial temperature (T) is 15.0°C. The final temperature (Tf) of the water is what we need to find.

We can set up the equation as follows:

Q (heat lost by the coin) = Q (heat gained by the water)

m (coin) * c (silver) * ΔT (coin) = m (water) * c (water) * ΔT (water)

14.0 g * 0.233 J/g°C * 0°C = 29.5 g * 4.18 J/g°C * (Tf - 15.0°C)

Simplifying the equation:

0 Joules = 122.85 J/°C * (Tf - 15.0°C)

0 = 122.85 Tf - 1842.75

Rearranging the equation to solve for Tf:

122.85 Tf = 1842.75

Tf = 1842.75 / 122.85

Tf = 15.0°C

Therefore, the final temperature of the water would be 15.0°C.

To find the final temperature of the water, we can use the principle of energy conservation. The heat gained by the water will be equal to the heat lost by the coin.

The heat gained by the water can be calculated using the formula:
Q_water = m_water * C_water * ΔT_water

Where:
Q_water = heat gained by water
m_water = mass of water
C_water = specific heat capacity of water
ΔT_water = change in temperature of water

The heat lost by the coin can be calculated using the formula:
Q_coin = m_coin * C_coin * ΔT_coin

Where:
Q_coin = heat lost by coin
m_coin = mass of coin
C_coin = specific heat capacity of coin (either nickel or silver)
ΔT_coin = change in temperature of coin

Since the coin was initially heated to 100°C and then added to water at 15°C, the change in temperature of the coin (ΔT_coin) will be the final temperature of the coin.

Setting the heat gained by the water equal to the heat lost by the coin, we have:
Q_water = Q_coin

m_water * C_water * ΔT_water = m_coin * C_coin * ΔT_coin

Substituting the given values:
(29.5 g) * (4.18 J/g°C) * (ΔT_water) = (14.0 g) * (0.233 J/g°C) * (ΔT_coin)

Simplifying the equation:
(29.5 g) * (4.18 J/g°C) * (ΔT_water) = (14.0 g) * (0.233 J/g°C) * (ΔT_coin)

We can solve this equation to find the value of ΔT_water, which will be the final temperature of the water.

ΔT_water = (14.0 g) * (0.233 J/g°C) * (ΔT_coin) / (29.5 g) * (4.18 J/g°C)

Calculating the value of ΔT_water using this equation will give us the final temperature of the water when the coin is made of silver.