Having trouble with this heat transfer problem

[xnitexlitex]

Homework Statement

When an immersion glass thermometer is used to measure the temperature of a liquid, the temperature reading will be affected by an error due to heat transfer between the liquid and the thermometer. Suppose you want to measure the temperature of 10 mL of water in a Pyrex glass vial thermally insulated from the environment. The empty vial has a mass of 5.0 g. The thermometer you use is made of Pyrex glass as well and has a mass of 18 g, of which 7 g is the mercury inside the thermometer. The thermometer is initially at room temperature (20.0°C). You place the thermometer in the water in the vial and, after a while, you read an equilibrium temperature of 27°C. What was the actual temperature of the water in the vial before the temperature was measured? The specific heat capacity of Pyrex glass around room temperature is 800 J/(kg K) and that of liquid mercury at room temperature is 140 J/(kg K).

Homework Equations

Q_a = Q_b
Q=mcΔT

The Attempt at a Solution

I've tried using Q_water+glass = Q_{thermometer+mercury} and Q_water = Q_pyrex + Q_mercury. I think my problem is finding the ΔT's. I have ΔT = 7 for the pyrex and the mercury, but I can't determine where my T_i should go for the water. are my ΔT's for the myrex and mercury right?

 

[technician]

I think the ΔT is 7 degrees for the glass thermometer+mercury.
You can calculate the heat energy needed by the glass thermometer and the mercury.
This heat energy came from the water and the glass vial.
Does this help?

 

[xnitexlitex]

I think the ΔT is 7 degrees for the glass thermometer+mercury.
You can calculate the heat energy needed by the glass thermometer and the mercury.
This heat energy came from the water and the glass vial.
Does this help?

Could you go into more detail, with equations if you can?