Extra power needed uphill for weight increase How much does weight mean when going uphill?

Let's say I'm going up a mountain with 1000 m of elevation and it's quite steep: 10%. Let's also say I can maintain 250 watts. If I add 1 kg of weight to the bike, can I determine how much slower (in time) I will be? How many seconds will I lose?

Please show me the formula and calculation of this, and feel free to elaborate on the whole subject.
Presuming you are doing a standing start and coming to a complete stop at the top of the hill.  The simple requirement is you need energy to move your from the bottom to the top.  Most of the energy required will be to raise potential energy of the payload (you and the bike).  Essentially you will be creating kinetic energy (moving the bike) by converting chemical energy in your body.  There will be loses due to heat, friction with the road surface and air resistance.  

Disregarding them for the moment (they are not negligible, but complicate the calculation).  

Potential Energy (PE) = m * g * h

Where:

m = mass 

g = gravitational acceleration 

h = height

PE is proportional to m, so a 10% increase in mass will increase the PE by 10%.  Meaning you will need 10% more kinetic energy to get to the top of the same hill. 

Power is work done (energy) divided by time:

P = W / t

Where:

P = Power in watts

W = work done or energy in Joules

t = time to do work.

If your power is constant we can rearrange the equation to get

P = (m * g * h)/ t

becomes:

t = (m * g * h) / P

so with constant power, gravity and hill height your time will increase proportional to increase in mass, given by the above equation.

If there is no wind air resistance will become less relevant the slower you are going.  Friction will increase due to increase in weight.  The steepness of the hill is theoretically irrelevant in this calculation.  You are gaining the same amount of Gravitation Potential energy when you the same mass to the same height.  So it shouldn't theoretically matter whether the hill is 10% or double as long and 5%.

However, as you are generating the energy you need to create it from chemical energy, and there is only so much you can generate at one time.  Your muscles will become inefficient and therefore on steeper hills you may require more energy than less steep ones.  So on a steeper hill wind resistance might become less relevant but power (how much energy you can put out over time) to weight ratio will become the most relevant factor.

What I am trying to point out in the last paragraph is that the energy you require to put into your body and for your body to convert into forward motion is not the same as the simple kinetic energy required to get you to the top of the hill.  However all things being equal a change in mass will have the same affect on the time as I stated in the equations.